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# 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) $$x^2 -3x - 10 = 0$$
(Splitting -3x as 2x - 5x)
$$\Rightarrow x^2 + 2x - 5x - 10 = 0$$
$$\Rightarrow x(x + 2) - 5(x + 2) = 0$$
$$\Rightarrow (x - 5)(x + 2) = 0$$

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

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

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

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)
$$\Rightarrow 100 x^2 – 10x - 10x + 1 = 0$$
$$\Rightarrow 10x(10x - 1) - 1(10x - 1) = 0$$
$$\Rightarrow (10x - 1)(10x - 1) = 0$$
$$\Rightarrow (10x - 1)^2 = 0$$

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

which is,
$$10x - 1 = 0$$
Thus, $$x = \frac{1}{10}$$