R.D. Sharma solutions for Mathematics [English] Class 10 chapter 4 - Quadratic Equations [1990 edition]

Check whether the following is quadratic equation or not.

16x² – 3 = (2x + 5) (5x – 3)

Check whether the following is quadratic equation or not.

`sqrt(3x^2) - 2x + 1/2 = 0`

Check whether the following is quadratic equation or not.

`x^2 + 1/x^2 = 5`

Check whether the following is quadratic equation or not.

`x - 3/x = x^2`

Check whether the following is quadratic equation or not.

`2x^2 - sqrt(3x) + 9 = 0`

Check whether the following is quadratic equation or not.

`x^2 - 2x - sqrtx - 5 = 0` 

Check whether the following is quadratic equation or not.

(x + 2)³ = x³ – 4

Check whether the following is quadratic equation or not.

`x + 1/x = 1`

Check whether the following is quadratic equation or not.

`x + 1/x = x^2`, x ≠ 0

Check whether the following is quadratic equation or not.

`(x + 1/x)^2 = 3(x + 1/x) + 4`

In the following, determine whether the given values are solutions of the given equation or not:

`x^2 - 3sqrt3x + 6 = 0, x = sqrt3, x = -2sqrt3`

In the following, determine whether the given values are solutions of the given equation or not:

`x + 1/x = 13/6, x = 5/6, x = 4/3`

In the following, determine whether the given values are solutions of the given equation or not:

2x² – x + 9 = x² + 4x + 3, x = 2, x = 3

In the following, determine whether the given values are solutions of the given equation or not:

`x^2 - sqrt2x - 4 = 0, x = -sqrt2, x = -2sqrt2`

In the following, determine whether the given values are solutions of the given equation or not:

`a^2x^2 - 3abx + 2b^2 = 0, x = a/b, x = b/a`

In the following, determine whether the given values are solutions of the given equation or not:

`sqrt(x^2 - 4x + 3) + sqrt(x^2 - 9) = sqrt(4x^2 - 14x + 16), x = 3`

In the following, find the value of k for which the given value is a solution of the given equation:

`7x^2 + kx - 3 = 0, x = 2/3`

In the following, find the value of k for which the given value is a solution of the given equation:

x² – x(a + b) + k = 0, x = a

In the following, find the value of k for which the given value is a solution of the given equation:

`kx^2 + sqrt2x - 4 = 0, x = sqrt2`

In the following, find the value of k for which the given value is a solution of the given equation:

x² + 3ax + k = 0, x = –a

If `x = 2/3` and x = –3 are the roots of the equation ax² + 7x + b = 0, find the values of a and b.

The product of two consecutive positive integers is 306. Form the quadratic equation to find the integers, if x denotes the smaller integer.

John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. Form the quadratic equation to find how many marbles they had to start with, if John had x marbles.

The height of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, form the quadratic equation to find the base of the triangle.

A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of articles produced in a day. On a particular day, the total cost of production was Rs. 750. If x denotes the number of toys produced that day, form the quadratic equation of find x. 

An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore. If the average speed of the express train is 11 km/hr more than that of the passenger train, form the quadratic equation to find the average speed of express train.

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Solve the following quadratic equations by factorization:

25x(x + 1) = – 4

Solve the following quadratic equations by factorization:

\[16x - \frac{10}{x} = 27\]

Solve the following quadratic equations by factorization:

6x² + 11x + 3 = 0

Solve the following quadratic equations by factorization:

\[2x^2 + ax - a^2 = 0\]

Solve the following quadratic equations by factorization:

`1/(x - 1) - 1/(x + 5) = 6/7, x ≠ 1, -5`

Solve the following quadratic equations by factorization:

`1/(x + 4) - 1/(x - 7) = 11/30, x ≠ 4, 7`

Solve the following quadratic equations by factorization:

\[\frac{1}{x - 3} + \frac{2}{x - 2} = \frac{8}{x}; x \neq 0, 2, 3\]

Solve the following quadratic equations by factorization:

\[\frac{16}{x} - 1 = \frac{15}{x + 1}; x \neq 0, - 1\]

Solve the following quadratic equations by factorization:

`(x + 3)/(x + 2) = (3x - 7)/(2x - 3); x ≠ -2, 3/2`

Solve the following quadratic equations by factorization:

`(x + 3)/(x - 2) - (1 - x)/x = 17/4; x ≠ 0, 2`

Solve the following quadratic equations by factorization:

`(x - 3)/(x + 3) - (x + 3)/(x - 3) = 48/7; x ≠ 3, -3`

Solve the following quadratic equations by factorization:

\[\frac{4}{x} - 3 = \frac{5}{2x + 3}, x \neq 0, - \frac{3}{2}\]

Solve the following quadratic equations by factorization:

\[\frac{3}{x + 1} - \frac{1}{2} = \frac{2}{3x - 1}, x \neq - 1, \frac{1}{3}\]

Solve the following quadratic equations by factorization:

\[\frac{3}{x + 1} + \frac{4}{x - 1} = \frac{29}{4x - 1}; x \neq 1, -1, \frac{1}{4}\]

Solve the following quadratic equations by factorization:

\[\frac{2}{x + 1} + \frac{3}{2(x - 2)} = \frac{23}{5x}; x \neq 0, -1, 2\]

Solve the following quadratic equations by factorization:

a²x² – 3abx + 2b² = 0

Solve the following quadratic equations by factorization:

\[9x^2 - 6 b^2 x - \left(a^4 - b^4 \right) = 0\]

Solve the following quadratic equations by factorization:

4x² + 4bx – (a² – b²) = 0

Solve the following quadratic equations by factorization:

`x^2 + (a + 1/a)x + 1 = 0`

Solve the following quadratic equations by factorization:

abx² + (b² – ac)x – bc = 0

Solve the following quadratic equations by factorization:

`3sqrt(5)x^2 + 25x - 10sqrt(5) = 0`

Solve the following quadratic equations by factorization:

\[\sqrt{3}x^2 - 2\sqrt{2}x - 2\sqrt{3} = 0\]

Solve the following quadratic equations by factorization:

`3x^2 - 2sqrt6x + 2 = 0`

Find the roots of the quadratic equation \[\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0\].

Solve the following quadratic equations by factorization:

\[3\left( \frac{7x + 1}{5x - 3} \right) - 4\left( \frac{5x - 3}{7x + 1} \right) = 11; x \neq \frac{3}{5}, - \frac{1}{7}\]

Solve the following quadratic equations by factorization:

\[\frac{x + 1}{x - 1} + \frac{x - 2}{x + 2} = 4 - \frac{2x + 3}{x - 2}; x \neq 1, -2, 2\]

Solve the following quadratic equation:

`a/((x - b)) + b/((x - a)) = 2, x ≠ b, a`

Solve the following quadratic equations by factorization:

`a/(x - a) + b/(x - b) = (2c)/(x - c)`

Solve the following quadratic equations by factorization:

\[\frac{1}{2a + b + 2x} = \frac{1}{2a} + \frac{1}{b} + \frac{1}{2x}\]

Solve the following quadratic equations by factorization:

`(x - 5)(x - 6) = 25/(24)^2`

Solve the following quadratic equations by factorization:

`7x + 3/x = 35 3/5`

Write the discriminant of the following quadratic equations:

(x – 1) (2x – 1) = 0

Write the discriminant of the following quadratic equations:

x² – 2x + k = 0, k ∈ R

Write the discriminant of the following quadratic equations:

`sqrt3x^2 + 2sqrt2x - 2sqrt3 = 0`

Write the discriminant of the quadratic equation (x + 5)² = 2 (5x − 3).

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

`3x^2 + 2sqrt5x - 5 = 0`

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

`2x^2 + 5sqrt3x + 6 = 0`

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

`sqrt2x^2 + 7x + 5sqrt2 = 0`

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

`2x^2 - 2sqrt2x + 1 = 0`

Solve for x:

`(x - 1)/(x - 2) + (x - 3)/(x - 4) = 3 1/3; x ≠ 2, 4`

Solve for x:

\[\frac{1}{x - 3} - \frac{1}{x + 5} = \frac{1}{6}, x \neq 3, - 5\]

Find the nature of the roots of the following quadratic equation. If the real roots exist, find them:

2x² – 3x + 5 = 0

Find the nature of the roots of the following quadratic equation. If the real roots exist, find them:

2x² – 6x + 3 = 0

Find the nature of the roots of the following quadratic equation. If the real roots exist, find them:

`3x^2 - 4sqrt3x + 4 = 0`

Determine the nature of the roots of the following quadratic equation:

`4x^2 + 4sqrt(3)x + 3 = 0`

Find the values of k for which the roots are real and equal in the following quadratic equation:

4x² – 2(k + 1)x + (k + 4) = 0

Find the values of k for which the roots are real and equal in the following quadratic equation:

4x² – 2(k + 1)x + (k + 1) = 0

Find the value of k for which the following equations have real and equal roots:

\[x^2 - 2\left( k + 1 \right)x + k^2 = 0\]

Find the values of k for which the roots are real and equal in the following quadratic equation:

k²x² – 2(2k – 1)x + 4 = 0

Find the value of k for which the following equations have real and equal roots:

\[\left( k + 1 \right) x^2 - 2\left( k - 1 \right)x + 1 = 0\]

Find the value of k for which the following equations have real and equal roots:

\[x^2 + k\left(2x + k - 1 \right) + 2 = 0\]

Find the values of k for which the roots are real and equal in the following equation:

2x² + kx + 3 = 0

Find the values of k for which the roots are real and equal in the following equation:

kx(x – 2) + 6 = 0

Find the values of k for which the roots are real and equal in the following equation:

x² – 4kx + k = 0

Find the values of k for which the roots are real and equal in the following equation:

\[kx\left( x - 2\sqrt{5} \right) + 10 = 0\]

Find the values of k for which the roots are real and equal in the following equation:

\[kx(x - 3) + 9 = 0\]

Find the values of k for which the roots are real and equal in the following equation:

\[4x^2 + kx + 3 = 0\]

Find the values of k for which the given quadratic equation has real and distinct roots:

kx² + 2x + 1 = 0

Find the values of k for which the given quadratic equation has real and distinct roots:

kx² + 6x + 1 = 0

For what value of k, (4 – k)x² + (2k + 4)x + (8k + 1) = 0, is a perfect square.

Find the values of k for which the quadratic equation \[\left( 3k + 1 \right) x^2 + 2\left( k + 1 \right)x + 1 = 0\] has equal roots. Also, find the roots.

Write all the values of k for which the quadratic equation x² + kx + 16 = 0 has equal roots. Find the roots of the equation so obtained.

Find the value of 'p' for which the equation px (x – 2) + 6 = 0 has two equal roots. Also, find the roots.

Find the values of p for which the quadratic equation \[\left( 2p + 1 \right) x^2 - \left( 7p + 2 \right)x + \left( 7p - 3 \right) = 0\] has equal roots. Also, find these roots.

Find that value of p for which the quadratic equation (p + 1)x² – 6(p + 1)x + 3(p + 9) = 0, p ≠ –1 has equal roots. Hence, find the roots of the equation.

Find the value(s) of p for which the quadratic equation given as (p + 4) x² – (p + 1) x + 1 = 0 has real and equal roots. Also, find the roots of the equation(s) so obtained.

Find the smallest value of p for which the quadratic equation x² − 2(p + 1)x + p² = 0 has real roots. Hence, find the roots of the equation so obtained.

Find the least positive value of k for which the equation x² + kx + 4 = 0 has real roots. 

If the roots of the equation (b – c) x² + (c – a) x + (a – b) = 0 are equal, then prove that 2b = a + c.

If the roots of the equation (a² + b²)x² – 2(ac + bd)x + (c² + d²) = 0 are equal, prove that `a/b = c/d`.

If the equation \[\left(1 + m^2 \right) x^2 + 2mcx + \left(c^2 - a^2 \right) = 0\] has equal roots, prove that c² = a²(1 + m²).

Determine the nature of the roots of the following quadratic equation:

(x – 2a)(x – 2b) = 4ab

Determine the nature of the roots of the following quadratic equation:

9a²b²x² – 24abcdx + 16c²d² = 0, a ≠ 0, b ≠ 0

Determine the nature of the roots of the following quadratic equation:

2(a² + b²)x² + 2(a + b)x + 1 = 0

Determine the nature of the roots of the following quadratic equation:

(b + c)x² – (a + b + c)x + a = 0

If the roots of the equations ax² + 2bx + c = 0 and `bx^2 - 2sqrt(ac)x + b = 0` are simultaneously real, then prove that b² = ac.

If p, q are real and p ≠ q, then show that the roots of the equation (p – q) x² + 5(p + q) x – 2(p – q) = 0 are real and unequal.

If the roots of the equation (c² – ab)x² – 2(a² – bc)x + b² – ac = 0 are real and equal, prove that either a = 0 or a³ + b³ + c³ = 3abc.

Show that the equation 2(a² + b²)x² + 2(a + b)x + 1 = 0 has no real roots, when a ≠ b.

Prove that both the roots of the equation (x – a)(x – b) + (x – b)(x – c) + (x – c)(x – a) = 0 are real but they are equal only when a = b = c.

If a, b, c are real numbers such that ac ≠ 0, then show that at least one of the equations ax² + bx + c = 0 and –ax² + bx + c = 0 has real roots.

The sum of two numbers is 8 and 15 times the sum of their reciprocals is also 8. Find the numbers.

Two numbers differ by 3 and their product is 504. Find the numbers.

The sum of two numbers a and b is 15, and the sum of their reciprocals `1/a` and `1/b` is `3/10`. Find the numbers a and b.

The sum of two numbers is 9. The sum of their reciprocals is `1/2`. Find the numbers.

Three consecutive positive integers are such that the sum of the square of the first and the product of other two is 46, find the integers.

The difference of squares of two numbers is 88. If the larger number is 5 less than twice the smaller number, then find the two numbers.

The sum of a numbers and its positive square root is `6/25`. Find the numbers.

The difference of two numbers is 4. If the difference of their reciprocals is `4/21`, find the numbers.

A two digit number is 4 times the sum of its digits and twice the product of its digits. Find the number.

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find two numbers.

The difference of two natural numbers is 3 and the difference of their reciprocals is \[\frac{3}{28}\]. Find the numbers.

The numerator of a fraction is 3 less than the denominator. If 2 is added to both the numerator and the denominator, then the sum of the new fraction and the original fraction is \[\frac{29}{20}\]. Find the original fraction.

Find a natural number whose square diminished by 84 is equal to thrice of 8 more than the given number.

A natural number, when increased by 12, equals 160 times its reciprocal. Find the number.

A 2-digit number is such that the product of its digits is 24. If 18 is subtracted from the number, the digits interchange their places. Find the number.

In a 2-digit number, the digit at the unit's place is 5 less than the digit at the ten's place. The product of the digits is 36. Find the number. 

The difference of squares of two positive integers is 400. Find the integers if twice of the smaller integer is 5 more than the greater integer.

A two digit number is such that the product of the digit is 12. When 36 is added to the number, the digits interchange their places. Find the numbers.

A 2 digit number is seven times the sum of its digits and two 2 more then 5 times the product of its digits. Find the number.

A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hour more, it would have taken 30 minutes less for a journey. Find the original speed of the train.

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Represent the following situation in the form of a quadratic equation.

A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.

A motorboat whose speed is 9 km/hr in still water, goes 15 km downstream and comes back in a total time of 3 hours 45 minutes. Find the speed of the stream. 

A train, travelling at a uniform speed for 360 km, would have taken 48 minutes less to travel the same distance if its speed were 5 km/h more. Find the original speed of the train.

A plane left 40 minutes late due to bad weather and in order to reach its destination, 1600 km away in time, it had to increase its speed by 400 km/hr from its usual speed. Find the usual speed of the plane.

A train travels at a certain average speed for a distance 63 km and then travels a distance of 72 km at an average speed of 6 km/hr more than the original speed, If it takes 3 hours to complete total journey, what is its original average speed?

An aeroplane left 50 minutes later than its scheduled time, and in order to reach the destination, 1250 km away, in time, it had to increase its speed by 250 km/hr from its usual speed. Find its usual speed.

A train travels at a certain average speed for a distanced of 54 km and then travels a distance of 63 km at an average speed of 6 km/hr more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?

The time taken by a person to travel an upward distance of 150 km was `2 1/2` hours more than the time taken in the downward return journey. If he returned at a speed of 10 km/h more than the speed while going up, find the speeds in each direction.

Ashu is x years old while his mother Mrs Veena is x² years old. Five years hence Mrs Veena will be three times old as Ashu. Find their present ages.

The sum of the ages of a man and his son is 45 years. Five years ago, the product of their ages was four times the man’s age at the time. Find their present ages.

The product of Shikha’s age five years ago and her age 8 years later is 30, her age at both times being given in years. Find her present age.

The product of Ramu’s age (in years) five years ago and his age (in years) nice years later is 15. Determine Ramu’s present age.

Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

A girls is twice as old as her sister. Four years hence, the product of their ages (in years) will be 160. Find their present ages.

The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is `1/3`. Find his present age.

If Zeba were younger by 5 years than what she really is, then the square of her age (in years) would have been 11 more than five times her actual age. What is her age now?

At present Asha’s age (in years) is 2 more than the square of her daughter Nisha’s age. When Nisha grows to her mother’s present age, Asha’s age would be one year less than 10 times the present age of Nisha. Find the present ages of both Asha and Nisha.

The age of a man is twice the square of the age of his son. Eight years hence, the age of the man will be 4 years more than three times the age of his son. Find the present age.

The hypotenuse of a right triangle is 25 cm. The difference between the lengths of the other two sides of the triangle is 5 cm. Find the lengths of these sides.

The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

The hypotenuse of a right triangle is `3sqrt10` cm. If the smaller leg is tripled and the longer leg doubled, new hypotenuse wll be `9sqrt5` cm. How long are the legs of the triangle?

A pole has to be erected at a point on the boundary of a circular park of diameter 13 meters in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 meters. Is it the possible to do so? If yes, at what distances from the two gates should the pole be erected?

Is it possible to design a rectangular mango grove whose length is twice its breadth and the area is 800 m²? If so, find its length and breadth.

Is it possible to design a rectangular park of perimeter 80 m and area 400 m²? If so, find its length and breadth.

The sum of the areas of two squares is 640 m². If the difference in their perimeter be 64 m, find the sides of the two squares.

The sum of the areas of two squares is 52 cm² and difference of their perimeters is 8 cm. Find the lengths of the sides of the two squares.

Represent the following situation in the form of a quadratic equation:

The area of a rectangular plot is 528 m². The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

In the centre of a rectangular lawn of dimensions 50 m × 40 m, a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be 1184 m² [see figure]. Find the length and breadth of the pond.

A takes 10 days less than the time taken by B to finish a piece of work. If both A and B together can finish the work in 12 days, find the time taken by B to finish the work.

If two pipes function simultaneously, a reservoir will be filled in 12 hours. One pipe fills the reservoir 10 hours faster than the other. How many hours will the second pipe take to fill the reservoir?

Two water taps together can fill a tank in `9 3/8` hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Two pipes running together can fill a tank in `11 1/9` minutes. If one pipe takes 5 minutes more than the other to fill the tank separately, find the time in which each pipe would fill the tank separately.

To fill a swimming pool two pipes are used. If the pipe of larger diameter used for 4 hours and the pipe of smaller diameter for 9 hours, only half of the pool can be filled. Find, how long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool?

Two water taps together can fill a tank in `1 7/8` hours. The tap with a longer diameter takes 2 hours less than the tap with a smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately.

A piece of cloth costs Rs. 35. If the piece were 4 m longer and each meter costs Rs. 1 less, the cost would remain unchanged. How long is the piece?

Some students planned a picnic. The budget for the food was Rs. 480. As eight of them failed to join the party, the cost of the food for each member increased by Rs. 10. Find how many students went for the picnic.

If the list price of a toy is reduced by Rs. 2, a person can buy 2 toys more for Rs. 360. Find the original price of the toy.

Rs. 9000 were divided equally among a certain number of persons. Had there been 20 more persons, each would have got Rs. 160 less. Find the original number of persons.

In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.

A student scored a total of 32 marks in class tests in Mathematics and Science. Had he scored 2 marks less in Science and 4 marks more in Mathematics, the product of his marks would have been 253. Find his marks in the two subjects.

A dealer sells an article for Rs. 24 and gains as much percent as the cost price of the article. Find the cost price of the article.

A pole has to be erected at a point on the boundary of a circular park of diameter 13 meters in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 meters. Is it the possible to do so? If yes, at what distances from the two gates should the pole be erected?

A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.

At t minutes past 2 pm, the time needed by the minutes hand of a clock to show 3 pm was found to be 3 minutes less than `t^2/4` minutes. Find t.

Find the value(s) of  'k' so that the quadratic equation 4x² + kx + 1 = 0 has real and equal roots.

Write a quadratic polynomial, sum of whose zeros is \[2\sqrt{3}\] and their product is 2.

Find the discriminant of the quadratic equation \[3\sqrt{3}x^2 + 10x + \sqrt{3} = 0\].

The sides of a right triangle are such that the longest side is 4 m more than the shortest side and the third side is 2 m less than the longest side. Find the length of each side of the triangle. Also, find the difference between the numerical values of the area and the perimeter of the given triangle.

If `x = (-1)/2`, is a solution of the quadratic equation 3x² + 2kx – 3 = 0, find the value of k. 

If x = 3 is one root of the quadratic equation x² – 2kx – 6 = 0, then find the value of k.

For what value of k, the roots of the equation x² + 4x + k = 0 are real?

Find the value of k for which the roots of the equation 3x² – 10x + k = 0 are reciprocal of each other.

Solve the quadratic equation 2x² – 5x – 1 = 0 for x.

Find the nature of the roots of the quadratic equation x² – 5x + 9 = 0.

Write a quadratic equation with roots –3 and 5.

Write the set of values of k for which the quadratic equation has 2x² + kx – 8 = 0 has real roots.

If \[1 + \sqrt{2}\] is a root of a quadratic equation with rational coefficients, write its other root.

Write the condition to be satisfied for which equations ax² + 2bx + c = 0 and \[bx^2 - 2\sqrt{ac}x + b = 0\] have equal roots.

If `(k + 1)x^2 + 3/2 x = 7` is a quadratic equation, then k cannot be equal to ______.

Every quadratic equation has exactly ______ roots.

The values of k for which the quadratic equation 2x² – kx + k = 0 has equal roots are ______.

If `1/2` is a root of the equation `x^2 + kx - 5/4 = 0`, then the value of k is ______.

If one root of the quadratic equation 3x² – 10x + k = 0 is reciprocal of the other, then k = ______.

The values of k for which the quadratic equation x² – 4kx + k = 0 has equal roots, are ______.

If the arithmetic mean of the roots of the equation x² – 6x + 8 = 0 is ______.

If the arithmetic mean of the roots of the equation x(x – 2) + 4аx = 5 is 3, then a = ______.

If the equation mx² + 2x + m = 0 is satisfied by only one real value of x, then the values of m are ______ and ______.

The quadratic equation with rational coefficients having `sqrt(3)/2` as a root is ______.

The total number of values of x satisfying x² – 3|x| + 2 = 0 is ______.

The number of real roots of the equation x² + 3x + 2 = 0 is ______.

If the coefficient of x² and the constant term of a quadratic equation have opposite signs, then the roots of the quadratic equation are ______ and ______.

If the coefficient of x² and the constant term of a quadratic equation have the same sign and if the coefficient of x term is zero, then the quadratic equation has ______ roots.