Quadratic Equation

NCERT Exercise 4.2

Part 1

Question 1: Find the roots of the following quadratic equations by factorization:

(i) `x^2 – 3x – 10 = 0`

Answer: `x^2 – 3x – 10 = 0`

Or, `x^2 – 5x + 2x – 10 = 0`

Or, `x(x – 5) + 2(x – 5) = 0`

Or, `(x + 2)(x – 5) = 0`

Now; case 1: `(x + 2) = 0`

Or, `x = - 2`

Case 2: `(x – 5) = 0`

Or, `x = 5`

Hence, roots are: - 2 and 5

(ii) `2x^2 + x – 6`

Answer: `2x^2 + x – 6 = 0`

Or, `2x^2 + 4x – 3x – 6 = 0`

Or, `2x(x + 2) – 3(x + 2) = 0`

Or, `(2x – 3)(x + 2) = 0`

Case 1: `(2x – 3) = 0`

Or, `2x = 3`

Or, `x = 3/2`

Case 2: `(x + 2) = 0`

Or, `x = - 2`

Hence, roots are – 2 and 3/2

(iii) `sqrt2x^2 + 7x + 5sqrt2 = 0`

Answer: `sqrt2 x^2 + 7x + 5sqrt2 = 0`

Or, `sqrt2 x^2 + 2x + 5x + 5sqrt2 = 0`

Or, `sqrt2x(x + sqrt2) + 5(x + sqrt2) = 0`

Or, `(sqrt2x + 5)(x + sqrt2) = 0`

Case 1: `(sqrt2x + 5) = 0`

Or, `sqrt2x = 5`

Or, `x=-(5)/(sqrt2)`

Case 2: `(x + sqrt2) = 0`

Or, `x = - sqrt2`

Hence, roots are;

`-5/sqrt2` and `-sqrt2`

Question (iv): `2x^2-x+1/8=0`

Answer:

`2x^2-x+1/8=0`

Or, `(16x^2-8x+1)/(8)=0`

Or, `16x^2-4x-4x+1=0`

Or, `16x(x-1/4)-4(x-1/4)=0`

Or, `(16x-4)(x-1/4)`

Hence, `16x-4=0`

Or, `16x=4`

Or, `x=(4)/(16)=1/4`

Similarly;

`x-1/4=0`

Or, `x=1/4`

Hence, ¼ is the root of this equation.

(v) `100x^2 – 20x + 1 = 0`

Answer: `100x^2 – 20x + 1 = 0`

Or, `100x^2 – 10x – 10x + 1 = 0`

Or, `10x(10x – 1) – 1(10x – 1) = 0`

Or, `(10x – 1)(10x – 1) = 0`

Hence; `x = (1)/(10)`