# SELINA Solutions for Class 10 Maths Chapter 5 - Quadratic Equations

Aim for top marks in the prelim and board exams with the Selina Solutions for ICSE Class 10 Mathematics Chapter 5 Quadratic Equations. Improve your analytical abilities to prove whether the given equation is a quadratic equation or not using our textbook solutions. Also, learn the concept of two equal roots and two distinct real roots through Maths practice.

Revise the quadratic formula and the factorisation method to prove quadratic equations with the support of our ICSE Class 10 Maths textbook solutions. To revisit the problem-solving methods, you can watch our concept videos or go through our well-written revision notes.

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## Chapter 5 - Quadratic Equations Exercise Ex. 5(A)

Question 1(iv)

x2 + 5x - 5 = (x - 3)2

Solution 1(iv)

x2 + 5x - 5 = (x - 3)2

x2 + 5x - 5 = x2 - 6x + 9

11x - 14 =0; which is not of the form ax2 + bx + c = 0.

Given equation is not a quadratic equation.

Question 1(v)

7x3 - 2x2 + 10 = (2x - 5)2

Solution 1(v)

7x3 - 2x2 + 10 = (2x - 5)2

7x3 - 2x2 + 10 = 4x2 - 20x + 25

7x3 - 6x2 + 20x - 15 = 0; which is not of the form ax2 + bx + c = 0.

Given equation is not a quadratic equation.

Question 1(vi)

(x - 1)2 + (x + 2)2 + 3(x +1) = 0

Solution 1(vi)

(x - 1)2 + (x + 2)2 + 3(x +1) = 0

x2 - 2x + 1 + x2 + 4x + 4 + 3x + 3 = 0

2x2 + 5x + 8 = 0; which is of the form ax2 + bx + c = 0.

Given equation is a quadratic equation.

Question 1(i)

Find which of the following equations are quadratic:

(3x - 1)2 = 5(x + 8)

Solution 1(i)

(3x - 1)2 = 5(x + 8)

(9x2 - 6x + 1) = 5x + 40

9x2 - 11x - 39 =0; which is of the form ax2 + bx + c = 0.

Given equation is a quadratic equation.

Question 1(ii)

5x2 - 8x = -3(7 - 2x)

Solution 1(ii)

5x2 - 8x = -3(7 - 2x)

5x2 - 8x = 6x - 21

5x2 - 14x + 21 =0; which is of the form ax2 + bx + c = 0.

Given equation is a quadratic equation.

Question 1(iii)

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

Solution 1(iii)

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

3x2 + x - 12x - 4 = 3x2 + 6x - x - 2

16x + 2 =0; which is not of the form ax2 + bx + c = 0.

Given equation is not a quadratic equation.

Question 2(i)

Is x = 5 a solution of the quadratic equation x2 - 2x - 15 = 0?

Solution 2(i)

x2 - 2x - 15 = 0

For x = 5 to be solution of the given quadratic equation it should satisfy the equation.

So, substituting x = 5 in the given equation, we get

L.H.S = (5)2 - 2(5) - 15

= 25 - 10 - 15

= 0

= R.H.S

Hence, x = 5 is a solution of the quadratic equation x2 - 2x - 15 = 0.

Question 2(ii)

Is x = -3 a solution of the quadratic equation 2x2 - 7x + 9 = 0?

Solution 2(ii)

2x2 - 7x + 9 = 0

For x = -3 to be solution of the given quadratic equation it should satisfy the equation

So, substituting x = 5 in the given equation, we get

L.H.S=2(-3)2 - 7(-3) + 9

= 18 + 21 + 9

= 48

R.H.S

Hence, x = -3 is not a solution of the quadratic equation 2x2 - 7x + 9 = 0.

Question 3

If  is a solution of equation 3x2 + mx + 2 = 0, find the value of m.

Solution 3

For x =  to be solution of the given quadratic equation it should satisfy the equation

So, substituting x =  in the given equation, we get

Question 4

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

Solution 4

For x =  and x = 1 to be solutions of the given quadratic equation it should satisfy the equation

So, substituting x =  and x = 1 in the given equation, we get

Solving equations (1) and (2) simultaneously,

Question 5

If 3 and -3 are the solutions of equation ax2 + bx - 9 = 0. Find the values of a and b.

Solution 5

For x = 3 and x = -3 to be solutions of the given quadratic equation it should satisfy the equation

So, substituting x = 3 and x = -3 in the given equation, we get

Solving equations (1) and (2) simultaneously,

## Chapter 5 - Quadratic Equations Exercise Ex. 5(B)

Question 1

Without solving, comment upon the nature of roots of each of the following equations :

(i)7x2 - 9x +2 =0 (ii)6x2 - 13x +4 =0

(iii)25x2 - 10x +1=0 (iv)

(v)x2 - ax - b2 =0 (vi)2x2 +8x +9=0

Solution 1

Question 2(i)

Find the value of p, if the following quadratic equation has equal roots : 4x2 - (p - 2)x + 1 = 0

Solution 2(i)

Question 2(ii)

Find the value of 'p', if the following quadratic equations have equal roots :

x2 + (p - 3)x + p = 0

Solution 2(ii)

x2 + (p - 3)x + p = 0

Here, a = 1, b = (p - 3), c = p

Since, the roots are equal,

b2- 4ac = 0

(p - 3)2- 4(1)(p) = 0

p2 + 9 - 6p - 4p = 0

p2- 10p + 9 = 0

p2-9p - p + 9 = 0

p(p - 9) - 1(p - 9) = 0

(p -9)(p - 1) = 0

p - 9 = 0 or p - 1 = 0

p = 9 or p = 1

Question 3

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

Solution 3

Question 4

Find the value of m, if the following equation has equal roots : (m - 2)x2 - (5+m)x +16 =0

Solution 4

Question 5

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

Solution 5

## Chapter 5 - Quadratic Equations Exercise Ex. 5(C)

Question 1

Solve :

Solution 1

Question 2

Solve :

Solution 2

Question 3

Solve :

Solution 3

Question 4

Solve :

Solution 4

Question 5

Solve :

Solution 5

Question 6

Solve :

Solution 6

Question 7

Solve :

Solution 7

Question 8

Solve :

Solution 8

Question 9

Solve :

Solution 9

Question 10

Solve :

Solution 10

Question 11

Solve :

Solution 11

Question 12

Solve :

Solution 12

Question 13

Solve :

Solution 13

Question 14

Solve :

Solution 14

Question 15

Solve :

Solution 15

Question 16

2x2 - 9x + 10 = 0, When

(i) x N

(ii) x Q

Solution 16

Question 17

Solve :

Solution 17

Question 18

Solve :

Solution 18

Question 19

Solve :

Solution 19

Question 20

Solve :

Solution 20

Question 21

Find the quadratic equation, whose solution set is :

(i) (ii)

Solution 21

Question 22(i)

Solution 22(i)

Question 22(ii)

Solution 22(ii)

Question 23

Find the value of x, if a + 1=0 and x2 + ax - 6 =0.

Solution 23

If a+1=0, then a = -1

Put this value in the given equation x2 + ax - 6 =0

Question 24

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

Solution 24

If a + 7 =0, then a = -7

and b + 10 =0, then b = - 10

Put these values of a and b in the given equation

Question 25

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

Solution 25

4(2x+3)2 - (2x+3) - 14 =0

Put 2x+3 = y

Question 26

Without solving the quadratic equation 6x2 - x - 2=0, find whether is a solution of this equation or not.

Solution 26

Consider the equation, 6x2 - x - 2=0

Put  in L.H.S.

Since L.H.S.= R.H.S., then is a solution of the given equation.

Question 27

Determine whether x = -1 is a root of the equation x2 - 3x +2=0

or not.

Solution 27

x2 - 3x +2=0

Put x = -1 in L.H.S.

L.H.S. = (-1)2 - 3(-1) +2

= 1 +3 +2=6 R.H.S.

Then x = -1 is not the solution of the given equation.

Question 28

If x = is a solution of the quadratic equation 7x2+mx - 3=0;

Find the value of m.

Solution 28

7x2+mx - 3=0

Given x = is the solution of the given equation.

Put given value of x in the given equation

Question 29

If x = -3 and x = are solutions of quadratic equation mx2 + 7x + n = 0, find the values of m and n.

Solution 29

Question 30

If quadratic equation x2 - (m + 1) x + 6=0 has one root as x =3;

find the value of m and the root of the equation.

Solution 30

Question 31

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.

Solution 31

Question 32

Solution 32

or x = -(a + b)

Question 33

Solution 33

Question 34

If -1 and 3 are the roots of x2+px+q=0
then find the values of p and q

Solution 34

## Chapter 5 - Quadratic Equations Exercise Ex. 5(D)

Question 1

Solve each of the following equations using the formula :

(i)x2 - 6x =27 (ii)x2 - 10x +21=0

(iii)x2 +6x - 10 =0 (iv)x2 +2x - 6=0

(v)3x2+ 2x - 1=0 (vi)2x2 + 7x +5 =0

(vii) (viii)

(ix) (x)

(xi) (xii)

(xiii) (xiv)

Solution 1

Question 2

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

(i)x2 - 8x+5=0

(ii)5x2 +10x - 3 =0

Solution 2

Question 3(iv)

x2 - 5x - 10 = 0

Solution 3(iv)

Question 3(iii)

Solve each of the following equations for x and give, in each case, your answer correct to two decimal places :

x2 - 3x - 9 =0

Solution 3(iii)

Question 3(ii)

Solve each of the following equations for x and give, in each case, your answer correct to two decimal places :

Solution 3(ii)

Question 3(i)

Solve each of the following equations for x and give, in each case, your answer correct to two decimal places :

(i)2x2 - 10x +5=0

Solution 3(i)

Question 4

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

(i)3x2 - 12x - 1 =0

(ii)x2 - 16 x +6= 0

(iii)2x2 + 11x + 4= 0

Solution 4

Question 5

Solve:

(i)x4 - 2x2 - 3 =0

(ii)x4 - 10x2 +9 =0

Solution 5

Question 6

Solve :

(i)(x2 - x)2 + 5(x2 - x)+ 4=0

(ii)(x2 - 3x)2 - 16(x2 - 3x) - 36 =0

Solution 6

Question 7

Solve :

(i)

(ii)

(iii)

Solution 7

Question 8

Solve the equation . Write your answer correct to two decimal places.

Solution 8

Question 9

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

Solution 9

Consider the given equation:

Question 10

(x - 1)2 - 3x + 4 = 0

Solution 10

Question 11

Solve the quadratic equation x2 - 3(x + 3) = 0; Give your answer correct to two significant figures.

Solution 11

x2 - 3(x + 3) = 0

## Chapter 5 - Quadratic Equations Exercise Ex. 5(E)

Question 1

Solve:

Solution 1

Question 2

Solve: (2x+3)2=81

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solve each of the following equations, giving answer upto two decimal places.(i)x2 - 5x -10=0(ii) 3x2 - x - 7 =0

Solution 12

Question 13

Solution 13

Question 14

Solve :

(i)x2 - 11x - 12 =0; when x N

(ii)x2 - 4x - 12 =0; when x I

(iii)2x2 - 9x + 10 =0; when x Q

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

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

Solution 20

Consider the given equation:

## Chapter 5 - Quadratic Equations Exercise Ex. 5(F)

Question 1(i)

Solve : (x+5)(x-5)=24

Solution 1(i)

Given: (x+5)(x-5)=24

Question 1(ii)

Solve :

Solution 1(ii)

Given:

Question 1(iii)

Solve :

Solution 1(iii)

Given:

or

Question 2

One root of the quadratic equation  is . Find the value of m. Also, find the other root of the equation.

Solution 2

Given quadratic equation is  …. (i)

One of the roots of (i) is , so it satisfies (i)

So, the equation (i) becomes

Hence, the other root is.

Question 3

One root of the quadratic equation  is -3, find its other root.

Solution 3

Given quadratic equation is  …. (i)

One of the roots of (i) is -3, so it satisfies (i)

Hence, the other root is 2a.

Question 4

If and ;find the values of x.

Solution 4

Given  i.e

So, the given quadratic equation becomes

Hence, the values of x are  and.

Question 5

Find the solution of the equation; if and .

Solution 5

Given quadratic equation is ….. (i)

Also, given and

and

So, the equation (i) becomes

Hence, the solution of given quadratic equation are  and.

Question 6

If m and n are roots of the equation where x ≠ 0 and x ≠ 2; find m × n.

Solution 6

Since, m and n are roots of the equation, we have

and

Hence, .

Question 7

Solve, using formula :

Solution 7

x = a + 1 or x = -a - 2 = -(a + 2)

Question 8

(i) When (integers)

(ii) When (rational numbers)

Solution 8

(i) When the equation  has no roots

(ii) When the roots of  are

or

Question 9

Find the value of m for which the equation  has real and equal roots.

Solution 9

The quadratic equation has real and equal roots if its discriminant is zero.

or

Question 10

Find the values of m for which equation  has equal roots. Also, find the roots of the given equation.

Solution 10

Given quadratic equation is …. (i)

The quadratic equation has equal roots if its discriminant is zero

When , equation (i) becomes

When , equation (i) becomes

x =

Question 11

Find the value of k for which equation  has real roots.

Solution 11

Given quadratic equation is …. (i)

The quadratic equation has real roots if its discriminant is greater than or equal to zero

Hence, the given quadratic equation has real roots for.

Question 12

Find, using quadratic formula, the roots of the following quadratic equations, if they exist

(i)

(ii)

Solution 12

D = b2 - 4ac = = 25 - 24 = 1

Since D > 0, the roots of the given quadratic equation are real and distinct.

or

D = b2 - 4ac = = 16 - 20 = - 4

Since D < 0, the roots of the given quadratic equation does not exist.

Question 13

Solve :

(i)  and x > 0.

(ii)  and x < 0.

Solution 13

or

But as x > 0, so x can't be negative.

Hence, x = 6.

or

But as x < 0, so x can't be positive.

Hence,