Selina solutions for Concise Mathematics [English] Class 10 ICSE chapter 5 - Quadratic Equations [Latest edition]

4x² – 9 = 0 implies x is equal to ______.

(x – 3) (x + 5) = 0 gives x equal to ______.

If 4 is a root of equation x² + kx – 4 = 0; the value of k is ______.

The equation 2x² – 3x + k = 0 is satisfied by x = 2; the value of k is ______.

If x² – 7x = 0; the value of x is ______.

If `sqrt (2/3)` is a solution of equation 3x² + mx + 2 = 0, find the value of m.

`2/3`and 1 are the solutions of equation mx² + nx + 6 = 0. Find the values of m and n.

The roots of the quadratic equation x² – 6x – 7 = 0 are ______.

The roots of quadratic equation x(x + 8) + 12 = 0 are ______.

If one root of equation (p – 3) x² + x + p = 0 is 2, the value of p is ______.

If `x + 1/x = 2.5`, the value of x is ______.

For quadratic equation `2x + 5/x = 5` :

Solve equation using factorisation method:

(2x – 3)² = 49

Solve equation using factorisation method:

(x + 1)(2x + 8) = (x + 7)(x + 3)

Solve equation using factorisation method:

4(2x – 3)² – (2x – 3) – 14 = 0

Solve equation using factorisation method:

2x² – 9x + 10 = 0, when:

1. x ∈ N
2. x ∈ Q

Solve equation using factorisation method:

`(x - 3)/(x + 3) + (x + 3)/(x - 3) = 2 1/2`

Solve equation using factorisation method:

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

Solve the following quadratic equations by factorization: 

`5/(x - 2) - 3/(x + 6) = 4/x`

Solve the following quadratic equations by factorization: 

`(1 + 1/(x + 1))(1 - 1/(x - 1)) = 7/8`

Find the quadratic equation, whose solution set is: 

{3, 5} 

Find the quadratic equation, whose solution set is: 

{−2, 3}

Solve:

`x/3 + 3/(6 - x) = (2(6 +x))/15; (x ≠ 6)`

Solve the equation `9x^2 + (3x)/4 + 2 = 0`, if possible, for real values of x.

Find the value of x, if a + 7 = 0; b + 10 = 0 and 12x² = ax – b.

Use the substitution y = 2x + 3 to solve for x, if 4(2x + 3)² – (2x + 3) – 14 = 0.

Solve:

`x/a - (a + b)/x = (b(a + b))/(ax)`

Solve:

`(1200/x + 2)(x - 10) - 1200 = 60`

Solve the equation:`14/(x+3)-1=5/(x+1); xne-3,-1` , for x

Solve the following quadratic equations by factorization:

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

Solve the following equation by factorization:

`sqrt(2x + 9) = (13 - x)`

Sovle for x:

`2800/(x - 100) - 2800/x = 1/2`

Solve the following quadratic equation:

`3x^2 - 2sqrt(6)x + 2 = 0`

If x² – 3x + 2 = 0, values of x correct to one decimal place are ______.

If x² – 4x – 5 = 0, values of x correct to two decimal places are ______.

If x² – 8x – 9 = 0; values of x correct to one significant figure are ______.

If x² – 2x – 3 = 0; values of x correct to two significant figures are ______.

The value (values) of x satisfying the equation x² – 6x – 16 = 0 is ______.

Solve the following equation for x and give, in the following case, your answer correct to one decimal place:

x² – 8x + 5 = 0

Solve the following equation for x and give, in the following case, your answer correct to one decimal place:

5x² + 10x – 3 = 0

Solve the following equation for x and give, in the following case, your answer correct to 2 decimal places:

2x² – 10x + 5 = 0

Solve the following equation for x and give, in the following case, your answer correct to 2 decimal places:

`4x + 6/x + 13 = 0`

Solve the following equation for x and give, in the following case, your answer correct to 2 decimal places:

4x² – 5x – 3 = 0

Solve the following equation for x and give, in the following case, your answer correct to 3 decimal places:

3x² – 12x – 1 = 0

Solve the following equation for x and give, in the following case, your answer correct to 3 decimal places:

x² – 16x + 6 = 0

Solve the following equation for x and give, in the following case, your answer correct to 3 decimal places:

2x² + 11x + 4 = 0

Solve the following equation and give your answer correct to 3 significant figures:

5x² – 3x – 4 = 0

Solve for x using the quadratic formula. Write your answer correct to two significant figures:

(x – 1)² – 3x + 4 = 0

x = 3 is a solution of the quadratic equation (k + 2)x² − kx + 6 = 0, then other root is ______.

Equation 2x² – 3x + 1 = 0 has ______.

Which of the following equations has two real and distinct roots?

If the roots of equation x² − 6x + k = 0 are real and distinct, then value of k is ______.

If the roots of x² – px + 4 = 0 are equal, the value (values) of p is ______.

Which of the following equations has imaginary roots?

One root of equation 3x² – mx + 4 = 0 is 1, the value of m is ______.

Without solving, comment upon the nature of roots of the following equation:

7x² – 9x + 2 = 0

Without solving, comment upon the nature of roots of the following equation: 

6x² – 13x + 4 = 0

Without solving, comment upon the nature of roots of the following equation:

25x² − 10x + 1 = 0

Without solving, comment upon the nature of roots of the following equation: 

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

The equation 3x² – 12x + (n – 5) = 0 has equal roots. Find the value of n.

Find the value of ‘m’, if the following equation has equal roots:

(m – 2)x² – (5 + m)x + 16 = 0

Find the value of k for which the equation 3x² – 6x + k = 0 has distinct and real roots.

Given that 2 is a root of the equation 3x² – p(x + 1) = 0 and that the equation px² – qx + 9 = 0 has equal roots, find the values of p and q.

Find the root of the following equation.

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

Use quadratic formula to solve x² = 4x.

Use quadratic formula to solve:

`3y + 5/(16y) = 2`

From the following equation, find the value of constant 'k' so that the equation has real and equal roots.

kx(x − 2) + 6 = 0

From the following equation, find the value of constant 'k' so that the equation has real and equal roots.

(k+ 4)x² + (k + 1)x + 1 = 0

If x⁴ – 5x² + 4 = 0; the values of x are:

For equation `1/x + 1/(x - 5) = 3/10`; one value of x is ______.

Which of the following is correct for the equation `1/(x - 3) - 1/(x + 5) = 1`?

Solve:

2x⁴ − 5x² +3= 0, Take x² = y

Solve:

x⁴ – 2x² – 3 = 0

Solve:

(x² – x)² + 5(x² – x) + 4 = 0

Solve:

(x² – 3x)² – 16(x² – 3x) – 36 = 0

Solve:

`sqrt(x/(x - 3)) + sqrt((x - 3)/x) = 5/2`

Solve: 

`((2x -3)/(x - 1)) - 4((x - 1)/(2x - 3)) = 3`

Solve: 

`((3x + 1)/(x + 1)) + ((x + 1)/(3x + 1)) = 5/2`

Solve:

`9(x^2 + 1/x^2) - 9(x + 1/x) - 52 = 0`

Solve:

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

Solve:

(x² + 5x + 4)(x² + 5x + 6) = 120

Solve the following quadratic equation:

`3((3x - 1)/(2x + 3)) - 2((2x + 3)/(3x - 1)) = 5, x ≠ 1/3, -3/2` 

Solve:

5^{x + 1} + 5^{2 − x} = 53 + 1. 

If (k + 2)x² − 2x + 1 = 0 has real roots then greatest value of k(∈ Z) is ______.

Find the greatest value of k ∈ N for which the equation x² − 4x + k = 0 has distinct real roots.

If the quadratic equation kx² + kx + 1 = 0 has real and distinct roots, the value of k is ______.

If x² – 4x = 5, the value of x is ______.

If x² – 7x = 0; the value of x is ______.

If x = 1 is a root of the equation `sqrtx + kx − 2` = 0; the value of k is ______.

The equation `sqrt(15 - 2x) = x`.

Assertion (A): x = 3. 

Reason (R): `sqrt(15 - 2x) = x` ⇒ 15 − 2x = x²

⇒ x² + 2x − 15 = 0 

⇒ x = −5 or x = 3

A quadratic equation 2x² + 5x − 3 = 0. 

Assertion (A): The roots of the equation 2x² + 5x − 3 = 0 are real and unequal.

Reason (R): For the equation ax² + bx +c = 0, the roots are real and unequal if b² − 4ac > 0.

One root of a quadratic equation is 3 + `sqrt2`.

Statement (1): The other root of the given quadratic equation is 3 − `sqrt2`.

Statement (2): If one root of the given quadratic equation is in the form of a surd, the other root is its conjugate.

If p – 15 = 0 and 2x² + px + 25 = 0; find the values of x.

Solve:

`1/p + 1/q + 1/x = 1/(x + p + q)`

Solve the quadratic equation 8x² – 14x + 3 = 0

1. When x ∈ I (integers)
2. When x ∈ Q (rational numbers)

Solve, using formula:

x² + x – (a + 2)(a + 1) = 0

If m and n are roots of the equation `1/x - 1/(x - 2) = 3`; where x ≠ 0 and x ≠ 2; find m × n.

One root of the quadratic equation 8x² + mx + 15 = 0 is `3/4`. Find the value of m. Also, find the other root of the equation.

Show that one root of the quadratic equation x² + (3 – 2a)x – 6a = 0 is –3. Hence, find its other root.

Find the solution of the equation 2x² – mx – 25n = 0; if m + 5 = 0 and n – 1 = 0.

Solve:

(a + b)²x² – (a + b)x – 6 = 0; a + b ≠ 0

Without solving the following quadratic equation, find the value of 'm' for which the given equation has real and equal roots.

x² + 2(m – 1)x + (m + 5) = 0

Find the value of k for which equation 4x² + 8x – k = 0 has real roots.

If x = −2 is a root of the equation 3x² + 7x + p = 1, find the values of p. Now find the value of k so that the roots of the equation x² + k(4x + k − 1) + p = 0 are equal.

An Air India flight from Mumbai to Colombo was delayed by 60 minutes due to the emergency landing of the plane in Vizag as a passenger in the flight suddenly had a cardiac arrest. Now to reach the destination of 1800 km from Vizag to Colombo in time, so that passengers could catch their connecting flights, the speed of the plane was increased by 300 km/h than the usual speed.

Based on the above information, answer the following questions:

1. Taking the usual speed of plane as x km/h, form a quadratic equation for the situation described above. 
2. Find the nature of the roots of the quadratic equation. 
3. Find the usual speed of the plane.