Question 1

We will factorize each quadratic equation to find its roots.

Step 1 — Factorize $x^2 - 3x - 10 = 0$

We need to split the middle term. The product of 1 and -10 is -10. We need two numbers that multiply to -10. They must add to -3. The numbers are -5 and 2. $x^2 - 5x + 2x - 10 = 0$ We group the terms. $x(x - 5) + 2(x - 5) = 0$ We factor out the common term. $(x - 5)(x + 2) = 0$

Step 2 — Find the roots for (i)

We set each factor to zero. $x - 5 = 0 \quad \text{or} \quad x + 2 = 0$ We solve for $x$. $x = 5 \quad \text{or} \quad x = -2$

$\boxed{x = 5, -2}$

Step 3 — Factorize $2x^2 + x - 6 = 0$

We need to split the middle term. The product of 2 and -6 is -12. We need two numbers that multiply to -12. They must add to -1. The numbers are -4 and 3. $2x^2 - 4x + 3x - 6 = 0$ We group the terms. $2x(x - 2) + 3(x - 2) = 0$ We factor out the common term. $(x - 2)(2x + 3) = 0$

Step 4 — Find the roots for (ii)

We set each factor to zero. $x - 2 = 0 \quad \text{or} \quad 2x + 3 = 0$ We solve for $x$. $x = 2 \quad \text{or} \quad 2x = -3$ $x = 2 \quad \text{or} \quad x = -\frac{3}{2}$

$\boxed{x = 2, -\frac{3}{2}}$

Step 5 — Factorize $\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0$

We need to split the middle term. The product of $\sqrt{2}$ and $5\sqrt{2}$ is 10. We need two numbers that multiply to 10. They must add to 7. The numbers are 5 and 2. $\sqrt{2}x^2 + 5x + 2x + 5\sqrt{2} = 0$ We group the terms. $x(\sqrt{2}x + 5) + \sqrt{2}(\sqrt{2}x + 5) = 0$ We factor out the common term. $(\sqrt{2}x + 5)(x + \sqrt{2}) = 0$

Step 6 — Find the roots for (iii)

We set each factor to zero. $\sqrt{2}x + 5 = 0 \quad \text{or} \quad x + \sqrt{2} = 0$ We solve for $x$. $\sqrt{2}x = -5 \quad \text{or} \quad x = -\sqrt{2}$ $x = -\frac{5}{\sqrt{2}} \quad \text{or} \quad x = -\sqrt{2}$

$\boxed{x = -\frac{5}{\sqrt{2}}, -\sqrt{2}}$

Step 7 — Factorize $2x^2 - x + \frac{1}{8} = 0$

First, we clear the fraction. We multiply the entire equation by 8. $16x^2 - 8x + 1 = 0$ We need to split the middle term. The product of 16 and 1 is 16. We need two numbers that multiply to 16. They must add to -8. The numbers are -4 and -4. $16x^2 - 4x - 4x + 1 = 0$ We group the terms. $4x(4x - 1) - 1(4x - 1) = 0$ We factor out the common term. $(4x - 1)(4x - 1) = 0$

Step 8 — Find the roots for (iv)

We set each factor to zero. $4x - 1 = 0 \quad \text{or} \quad 4x - 1 = 0$ We solve for $x$. $4x = 1 \quad \text{or} \quad 4x = 1$ $x = \frac{1}{4} \quad \text{or} \quad x = \frac{1}{4}$

$\boxed{x = \frac{1}{4}, \frac{1}{4}}$

Step 9 — Factorize $100x^2 - 20x + 1 = 0$

We need to split the middle term. The product of 100 and 1 is 100. We need two numbers that multiply to 100. They must add to -20. The numbers are -10 and -10. $100x^2 - 10x - 10x + 1 = 0$ We group the terms. $10x(10x - 1) - 1(10x - 1) = 0$ We factor out the common term. $(10x - 1)(10x - 1) = 0$

Step 10 — Find the roots for (v)

We set each factor to zero. $10x - 1 = 0 \quad \text{or} \quad 10x - 1 = 0$ We solve for $x$. $10x = 1 \quad \text{or} \quad 10x = 1$ $x = \frac{1}{10} \quad \text{or} \quad x = \frac{1}{10}$

$\boxed{x = \frac{1}{10}, \frac{1}{10}}$

Answer

(i) The roots are 5 and -2. (ii) The roots are 2 and -3/2. (iii) The roots are -5/√2 and -√2. (iv) The roots are 1/4 and 1/4. (v) The roots are 1/10 and 1/10.