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

(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 x as 4x - 3x)
\(2x^2 +4x - 3x - 6 = 0\)
\( 2x(x + 2) - 3(x + 2) = 0 \)
\( (2x - 3)(x + 2) = 0 \)
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, \(x = \frac{3}{2}\) or \(x = -2\)

(iii)
\(\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0 \)
(Splitting 7x as 2x + 5x)
\(\sqrt{2}x^2 + 2x + 5x + 5\sqrt{2} = 0 \)
\( \sqrt{2}x(x + \sqrt{2}) + 5(x + \sqrt{2}) = 0 \)
\( (\sqrt{2}x + 5)(x + \sqrt{2}) = 0 \)
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, \(x = \frac{-5}{\sqrt{2}}\) or \(x = \sqrt{-2}\)

(iv)
\(2x^2 – x + \frac { 1 }{ 8 } = 0 \)
Multiplying and dividing the entire equation with 8, we get -
\(\frac{1}{8}(16x^2 - 8x + 1) = 0\)
(Splitting -8x as -4x - 4x)
\(\frac{1}{8}(16x^2 - 4x - 4x + 1) = 0\)
\( \frac{1}{8}[4x(4x - 1) - 1(4x - 1)] = 0 \)
\( \frac{1}{8}(4x - 1)(4x - 1) = 0 \)
\( \frac{1}{8}(4x - 1)^2= 0 \)
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, \(x = \frac{1}{4}\)

(v)
\(100 x^2 – 20x + 1 = 0\)
(Splitting -20x as - 10x - 10x)
\(100 x^2 – 10x - 10x + 1 = 0\)
\( 10x(10x - 1) - 1(10x - 1) = 0 \)
\( (10x - 1)(10x - 1) = 0 \)
\( (10x - 1)^2 = 0 \)
The roots of this equation are the values of x for which \( (10x - 1)^2 = 0 \)
which is,
\(10x - 1 = 0 \)
Thus, \(x = \frac{1}{10}\)