Q1(iii):

Find the roots of the following quadratic equations by factorisation: (iii) $\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0$

Given: A quadratic equation $\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0$.

To Find: The roots of the given quadratic equation using the method of factorisation.

Step 1: Identify the coefficients of the quadratic equation.

The standard form of a quadratic equation is $ax^2 + bx + c = 0$. Comparing our equation $\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0$ with the standard form, we have:

$a = \sqrt{2}$

$b = 7$

$c = 5\sqrt{2}$

Step 2: Determine the product $a \times c$ and find two numbers that sum to $b$.

We need to find two numbers whose product is $a \times c$ and whose sum is $b$.

Product = $a \times c = \sqrt{2} \times 5\sqrt{2} = 5 \times (\sqrt{2})^2 = 5 \times 2 = 10$.

Sum = $b = 7$.

We look for two numbers $p$ and $q$ such that $p \times q = 10$ and $p + q = 7$.

The factors of $10$ are $(1, 10)$ and $(2, 5)$. Since $2 + 5 = 7$, the required numbers are $2$ and $5$.

Step 3: Split the middle term.

Rewrite the middle term $7x$ as $(2x + 5x)$:

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

Step 4: Factor by grouping.

Group the terms to extract common factors:

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

Note that $2$ can be written as $\sqrt{2} \times \sqrt{2}$. Thus, $\sqrt{2}x^2 + (\sqrt{2} \times \sqrt{2})x = \sqrt{2}x(x + \sqrt{2})$.

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

Step 5: Factor out the common binomial $(x + \sqrt{2})$.

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

Step 6: Solve for $x$ by setting each factor to zero.

[By the Zero Product Property, if $A \times B = 0$, then $A = 0$ or $B = 0$]

Case 1: $x + \sqrt{2} = 0$

$x = -\sqrt{2}$

Case 2: $\sqrt{2}x + 5 = 0$

$\sqrt{2}x = -5$

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

To rationalise the denominator: $x = -\frac{5}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = -\frac{5\sqrt{2}}{2}$

Final Answer: The roots of the quadratic equation are $-\sqrt{2}$ and $-\frac{5\sqrt{2}}{2}$.