**Chapter 2 Quadratic Equations Set 2.5**

## Chapter 2 Quadratic Equations Set 2.5

**Question 1.Fill in the gaps and complete.**

Answer:

Answer:

**Question 2.Find the value of discriminant.i. x**

^{2}+ 7x – 1 = 0ii. 2y

^{2}– 5y + 10 = 0iii. √2 x

^{2}+ 4x + 2√2 = 0Solution:

i. x

^{2}+7 x – 1 = 0

Comparing the above equation with

ax

^{2}+ bx + c = 0, we get

a = 1, b = 7, c = -1

∴ b

^{2}– 4ac = (7)

^{2}– 4 × 1 × (-1)

= 49 + 4

∴ b

^{2}– 4ac = 53

ii. 2y^{2} – 5y + 10 = 0

Comparing the above equation with

ay^{2} + by + c = 0, we get

a = 2, b = -5, c = 10

∴ b^{2 }– 4ac = (-5)2 -4 × 2 × 10

= 25 – 80

∴ b^{2} – 4ac = -55

iii. √2 x^{2} + 4x + 2√2 = 0

Comparing the above equation with

ax + bx + c = 0, we get

a = √2,b = 4, c = 2√2

∴ b^{2} – 4ac = (4)2 – 4 × √2 × 2√2

= 16 – 16

∴ b^{2} – 4ac =0

**Question 3.Determine the nature of roots of the following quadratic equations.i. x**

^{2}– 4x + 4 = 0ii. 2y

^{2}– 7y + 2 = 0iii. m

^{2}+ 2m + 9 = 0Solution:

i. x

^{2}– 4x + 4= 0

Comparing the above equation with

ax

^{2}+ bx + c = 0, we get

a = 1,b = -4, c = 4

∴ ∆ = b

^{2}– 4ac

= (- 4)

^{2}– 4 × 1 × 4

= 16 – 16

∴ ∆ = 0

∴ Roots of the given quadratic equation are real and equal.

ii. 2y^{2} – 7y + 2 = 0

Comparing the above equation with

ay^{2} + by + c = 0, we get

a = 2, b = -7, c = 2

∴ ∆ = b^{2} – 4ac

= (- 7)^{2} – 4 × 2 × 2

= 49 – 16

∴ ∆ = 33

∴ ∆ > 0

∴ Roots of the given quadratic equation are real and unequal.

iii. m^{2} + 2m + 9 = 0

Comparing the above equation with

am^{2} + bm + c = 0, we get

a = 1,b = 2, c = 9

∴ ∆ = b^{2} – 4ac

= (2)^{2} – 4 × 1 × 9

= 4 – 36

∴ ∆ = -32

∴ ∆ < 0

∴ Roots of the given quadratic equation are not real.

**Question 4.Form the quadratic equation from the roots given below.i. 0 and 4ii. 3 and -10iv. 2 – √5, 2 + √5Solution:**

i. Let a = 0 and β = 4

∴ α + β = 0 + 4 = 4

and α × β = 0 × 4 = 0

∴ The required quadratic equation is

x

^{2}– (α + β) x + αβ = 0

∴ x

^{2}– 4x + 0 = 0

∴ x

^{2}– 4x = 0

ii. Let α = 3 and β = -10

∴ α + β = 3 – 10 = -7

and α × β = 3 × -10 = -30

∴ The required quadratic equation is

x^{2} – (α + β)x + αβ = 0

∴ x^{2} – (-7) x + (-30) = 0

∴ The required quadratic equation is

x^{2} – (α + β)x + αβ = 0

∴ x^{2} – 4x – 1 = 0

**Question 5.Sum of the roots of a quadratic equation is double their product. Find k if equation is x ^{2} – 4kx + k + 3 = 0.Solution:**

x

^{2}– 4kx + k + 3 = 0

Comparing the above equation with

ax

^{2}+ bx + c = 0, we get

a = 1, b = – 4k, c = k + 3

Let α and β be the roots of the given quadratic equation.

According to the given condition,

**Question 6.α, β are roots of y ^{2} – 2y – 7 = 0 find,i. α^{2} + β^{2}ii. α^{3} + β^{3}Solution:**

y

^{2}– 2y – 7 = 0

Comparing the above equation with

ay

^{2}+ by + c = 0, we get

a = 1, b = -2, c = -7

**Question 7.The roots of each of the following quadratic equations are real and equal, find k.i. 3y**

^{2}+ ky + 12 = 0ii. kx (x-2) + 6 = 0Solution:

i. 3y

^{2}+ kg + 12 = 0

Comparing the above equation with

ay

^{2}+ by + c = 0, we get

a = 3, b = k, c = 12

∴ ∆ = b

^{2}– 4ac

= (k)

^{2}– 4 × 3 × 12

= k

^{2}– 144 = k

^{2}– (12)

^{2}

∴ ∆ = (k + 12) (k – 12) …[∵ a

^{2}– b

^{2}= (a + b) (a – b)]

Since, the roots are real and equal.

∴ ∆ = 0

∴ (k + 12) (k – 12) = 0

∴ k + 12 = 0 or k – 12 = 0

∴ k = -12 or k = 12

ii. kx (x – 2) + 6 = 0

∴ kx^{2} – 2kx + 6 = 0

Comparing the above equation with

ax^{2} + bx + c = 0, we get

a = k, b = -2k, c = 6

∴ ∆ = b^{2} – 4ac

= (-2k)^{2} – 4 × k × 6

= 4k^{2} – 24k

∴ ∆ = 4k (k – 6)

Since, the roots are real and equal.

∴ ∆ = 0

∴ 4k (k – 6) = 0

∴ k(k – 6) = 0

∴ k = 0 or k – 6 = 0

But, if k = 0 then quadratic coefficient becomes zero.

∴ k ≠ 0

∴ k = 6

**Question 1.Fill in the blanks. (Textbook pg. no. 44)Solution:**

**Question 2.Determine nature of roots of the quadratic equation: x ^{2} + 2x – 9 = 0 (Textbook pg. no. 45)Solution:**

∴ The roots of the given equation are real and unequal.

**Question 3.Fill in the empty boxes properly. (Textbook pg. no. 46)Solution:**

10x

^{2}+ 10x + 1 = 0

Comparing the above equation with

ax

^{2}+ bx + c = 0, we get

a = 10, b = 10, c = 1

**Question 4.Write the quadratic equation if addition of the roots is 10 and product of the roots is 9. (Textbook pg, no. 48)Answer:**

**Question 5.What will be the quadratic equation if α = 2, β = 5. (Textbook pg. no, 48)Solution:**