Q1. Find the roots of the following Quadratic Equations by factorization.

(i) \(x^2 - 3x - 10 = 0\)

(ii) \(2x^2 + x - 6 = 0\)

(iii) \(\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0\)

(iv) \(2x^2 - x + {{1} \over 8} = 0\)

(v) \(100x^2 - 20x + 1 = 0\)

(i) \(x^2 - 3x - 10 = 0\)

(ii) \(2x^2 + x - 6 = 0\)

(iii) \(\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0\)

(iv) \(2x^2 - x + {{1} \over 8} = 0\)

(v) \(100x^2 - 20x + 1 = 0\)

** (i) ** \(x^2 -3x - 10 = 0\)

(Splitting * -3x * as * 2x - 5x*)

\( x^2 + 2x - 5x - 10 = 0 \)

\( x(x + 2) - 5(x + 2) = 0 \)

\( (x - 5)(x + 2) = 0 \)

The roots of this equation are the values of x for which \( (x - 5)(x + 2) = 0 \)

which are,

\( x - 5 = 0 \) or \( x + 2 = 0\)

Thus,** \(x = 5\) or \(x = -2\)
(ii)** \(2x^2 +x - 6 = 0\)

(Splitting

\(2x^2 +4x - 3x - 6 = 0\)

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

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

The roots of this equation are the values of x for which \( (2x - 3)(x + 2) = 0 \)

which are,

\( 2x - 3 = 0 \) or \( x + 2 = 0\)

Thus,

(iii)

(Splitting

\(\sqrt{2}x^2 + 2x + 5x + 5\sqrt{2} = 0 \)

\( \sqrt{2}x(x + \sqrt{2}) + 5(x + \sqrt{2}) = 0 \)

\( (\sqrt{2}x + 5)(x + \sqrt{2}) = 0 \)

The roots of this equation are the values of x for which \( (\sqrt{2}x + 5)(x + \sqrt{2}) = 0 \)

which are,

\(\sqrt{2}x + 5 = 0 \) or \( x + \sqrt{2} = 0\)

Thus,

(iv)

Multiplying and dividing the entire equation with 8, we get -

\(\frac{1}{8}(16x^2 - 8x + 1) = 0\)

(Splitting

\(\frac{1}{8}(16x^2 - 4x - 4x + 1) = 0\)

\( \frac{1}{8}[4x(4x - 1) - 1(4x - 1)] = 0 \)

\( \frac{1}{8}(4x - 1)(4x - 1) = 0 \)

\( \frac{1}{8}(4x - 1)^2= 0 \)

The roots of this equation are the values of x for which \( \frac{1}{8}(4x - 1)^2= 0 \)

which is,

\( 4x - 1 = 0\)

Thus,

(v)

(Splitting

\(100 x^2 – 10x - 10x + 1 = 0\)

\( 10x(10x - 1) - 1(10x - 1) = 0 \)

\( (10x - 1)(10x - 1) = 0 \)

\( (10x - 1)^2 = 0 \)

The roots of this equation are the values of x for which \( (10x - 1)^2 = 0 \)

which is,

\(10x - 1 = 0 \)

Thus, \(x = \frac{1}{10}\)