R.S. Aggarwal solutions for Mathematics [English] Class 10 ICSE chapter 6 - Problems on Quadratic Equations [Latest edition]

Find two natural numbers whose sum is 50 and product 525.

The difference of two natural numbers is 7 and their product is 450. Find the numbers.

Find two consecutive positive even integers whose product is 224.

Find two consecutive natural numbers, the sum of whose squares is 145.

The sum of two natural numbers is 12 and the sum of their squares is 74. Find the numbers.

Find two consecutive multiples of 3 whose product is 270.

The sum of a natural number and its reciprocal is $$\frac{65}{8}$$. Find the natural number.

The sum of two numbers is 2 and the sum of their reciprocals is 2.25. Find the numbers.

The sum of the squares of three consecutive odd numbers is 2531. Find the numbers.
[Hint : Let the required odd numbers be x, x + 2 and x + 4. Then,
$$x^2 + (x + 2)^2 + (x + 4)^2 = 2531 \Rightarrow x^2 + 4x - 837 = 0 \Rightarrow x^2 + 31x - 27x - 837 = 0$$]

The product of two consecutive natural numbers which are multiples of 3 is equal to 810. Find the two numbers. 

A number consists of two digits whose product is 18. If 27 is added to the number, the digits interchange their places. Find the number.

A two-digit number contains the smaller of the two digits in the unit place. The product of the digits is 40 and the difference between the digits is 3. Find the number.

The sum of the numerator and denominator of a certain fraction is 10. If 1 is subtracted from both the numerator and denominator, the fraction is decreased by $$\frac{2}{21}$$. Find the fraction.

Two years ago, a man’s age was three times the square of his son’s age. In three years time, his age will be four times his son’s age. Find their present ages.

Ashish goes to his friends house which is 12 km away from his house. He covers half of the distance at a speed of $$x$$ km per hour and the remaining at $$(x + 2)$$ km per hour. If he takes 2 hrs 30 min. to cover the whole distance, find the value of $$x$$.

By increasing the speed of a car by 10 km/hr, the time of journey for a distance of 72 km. is reduced by 36 minutes. Find the original speed of the car.

A train covers a distance of 600 km at $$x$$ km/hr. Had the speed been $$(x + 20)$$ km/hr, the time taken to cover the same distance would have been reduced by 5 hours. Write down an equation in $$x$$ and solve it to evaluate $$x$$.

A train covers a distance of 780 km at $$x$$ km/hr. Had the speed been $$(x - 5)$$ km/hr, the time taken to cover the same distance would have been increased by 1 hour. Write down an equation in $$x$$ and solve it to evaluate $$x$$.

The distance by road between two towns A and B is 216 km and by rail it is 208 km. A car travels at a speed of x km/hr and the train travels at a speed which is 16 km/hr faster than the car. Calculate:

1. the time taken by the car to reach town B from A, in terms of x;
2. the time taken by the train to reach town B from A, in terms of x.
3. If the train takes 2 hours less than the car, to reach town B, obtain an equation in x and solve it.
4. Hence, find the speed of the train.

Car A travels ‘x’ km for every litre of petrol, while car B travels (x + 5) km for every litre of petrol.

1. Write down the number of litres of petrol used by car A and car B in covering a distance of 400 km.
2. If car A uses 4 litres of petrol more than car B in covering 400 km. write down an equation, in A and solve it to determine the number of litres of petrol used by car B for the journey.

The speed of a boat in still water is $$x$$ km/hr and the speed of the stream is 3 km/hr. 

1. Write the speed of the boat upstream, in terms of $$x$$. 
2. Write the speed of the boat downstream, in terms of $$x$$. 
3. If the boat goes 15 km upstream and 22 km downstream in 5 hours, write an equation in $$x$$ to represent the statement. 
4. Solve the equation to evaluate $$x$$.

A booster pump can be used for filling as well as for emptying a tank. The capacity of the tank is $$2400 \text{ m}^3$$. The emptying capacity of the tank is $$10 \text{ m}^3$$ per minute higher than its filling capacity and the pump needs 8 minutes lesser to empty the tank than it needs to fill it. What is the filling capacity of the pump?
[Hint : Let the filling capacity of the pump be $$x \text{ m}^3/\text{min}$$.
Then, emptying capacity of the pump $$= (x + 10) \text{ m}^3/\text{min}$$.
$$\therefore \frac{2400}{x} - \frac{2400}{(x + 10)} = 8. \text{ Solve for } x.$$]

The hypotenuse of a right-angled triangle is 20 meters. If the difference between the lengths of the other sides be 4 meters, find the other sides. 

The lengths of the sides of a right triangle are (2x – 1) m, (4x) m and (4x + 1) m, where x > 0. Find :

1. the value of x, 
2. the area of the triangle.

Two squares have sides x cm and (x + 5) cm. The sum of their areas is 697 sq. cm.

1. Express this as an algebraic equation in x. 
2. Solve this equation to find the sides of the squares.

The area of a right-angled triangle is \[96\ \text{m}^{2}\]. If its base is three times its altitude, find the base.

The lengths of the parallel sides of a trapezium are (x + 8) cm and (2x + 3) cm, and the distance between them is (x + 4) cm. If its area is 590 cm², find the value of x.

The ratio between the length and the breadth of a rectangular field is 3 : 2. If only the length is increased by 5 metres, the new area of the field will be 2600 sq. metres. What is the breadth of the rectangular field?
[Hint : Let, length = (3x) metres and breadth = (2x) metres. Then, (3x + 5) × 2x = 2600.]

The perimeter of a rectangular plot of land is 114 metres and its area is 810 square metres.

1. Take the length of plot as x metres. Use the perimeter 114 m to write the value of the breadth in terms of x. 
2. Use the values of length, breadth and area to write an equation in x. 
3. Solve the equation to find the length and breadth of the plot.

A man buys an article for ₹ x and sells it for ₹ 56 at a gain of x%. Find the value of x.

[Hint : $$\frac{(100 + x)}{100} \times x = 56.$$]

₹ 6400 were divided equally among x persons. Had this money been divided equally among (x + 14) persons, each would have got ₹ 28 less. Find the value of x.

₹ 7500 were divided equally among a certain number of children. Had there been 20 less children, each would have received ₹ 100 more. Find the original number of children.

A shopkeeper buys x books for ₹ 720.

1. Write the cost of 1 book in terms of x. 
2. If the cost per book be ₹ 5 less, the number of books that could be bought for ₹ 720 would be 2 more. 

Write down the equation in x and solve it to find x.

A fruit-seller bought x apples for ₹ 1200.

1. Write the cost price of each apple in terms of x. 
2. If 10 of the apples were rotten and he sold each of the rest at ₹ 3 more than the cost price of each, write the selling price of (x – 10) apples. 
3. If he made a profit of ₹ 60 in this transaction, form an equation in x and solve it to evaluate x.

Some students planned a picnic. The budget for the food was ₹ 2400. As 8 of them failed to join the party, the cost of the food for each member increased by ₹ 50. Find how many students went to the picnic.

A bus covers a distance of 240 km at a uniform speed. Due to heavy rain its speed gets reduced by 10 km/h and as such it takes two hrs longer to cover the total distance. Assuming the uniform speed to be ‘x’ km/h, form an equation and solve it to evaluate ‘x’.

A man covers a distance of 100 km, travelling with a uniform speed of x km/hr. Had the speed been 5 km/hr more it would have taken 1 hour less. Find x the original speed.

If the sum of two natural numbers is 27 and their product is 182, then the smaller number is ______.

The sum of the squares of two consecutive odd natural numbers is 74. The greater number is ______.

Two natural numbers differ by 2 and the sum of their squares is 202. The sum of the numbers is ______.

₹40 is distributed between two friends such that the product of their shares is 364. The difference of their shares is ______.

The length of a rectangle is 4 cm more than its breadth. If the area of the rectangle is 96 cm², then the perimeter of the rectangle is ______.

If four times the area of a square is 484 cm², then perimeter of the square is ______.

Sum of the squares of the two consecutive positive integers is 365. The sum of the numbers is ______.

The altitude of a right triangle is 17 cm less than its base. If the hypotenuse is 25 cm, then the perimeter of the triangle is ______.

The cost of an article is ₹3 more than twice the total number of articles. If the cost of all the articles is ₹189, then the number of articles is ______.

The diagonal of a rectangular field is 60 m more than the shorter side. If the longer side is 30 m more than the shorter side, then the sides are ______.

Neha's father is 28 years older than her. The product of their ages (in years) 4 years ago was 245. If present age of Neha is x years, then the algebraic representation of this information in the form of quadratic equation is ______.

A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/hr less, then it would have taken 3 hours more to cover the same distance. If the initial speed of the train is x km/hr, then representation of this information algebraically is ______.

The number of litres of diesel used by car X is ______.

The number of litres of diesel used by car Y is ______.

If car X used 4 litres of diesel more than car Y in the journey, then :

The amount of diesel used by car X is ______.

The marked price of a toy is same as the percentage of GST that is charged. The price of the toy is ₹ 24 including GST. Taking the marked price as x, form an equation and solve it to find x.