Q1(i):

Find the roots of the following quadratic equations by factorisation: (i) $x^2 – 3x – 10 = 0$

Given: A quadratic equation $x^2 - 3x - 10 = 0$.

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

Step 1: Understanding the Factorisation Method (Splitting the Middle Term)
To factorise a quadratic equation of the form $ax^2 + bx + c = 0$, we look for two numbers, say $p$ and $q$, such that:
1. The product of the numbers is equal to the product of the coefficient of $x^2$ and the constant term: $p \times q = a \times c$.
2. The sum of the numbers is equal to the coefficient of $x$: $p + q = b$.
In the given equation $x^2 - 3x - 10 = 0$, we have $a = 1$, $b = -3$, and $c = -10$.
Therefore, we need two numbers such that:
Product ($p \times q$) = $1 \times (-10) = -10$
Sum ($p + q$) = $-3$

Step 2: Identifying the Factors
We list the pairs of factors of $-10$:
- $(1, -10) \implies 1 + (-10) = -9$
- $(-1, 10) \implies -1 + 10 = 9$
- $(2, -5) \implies 2 + (-5) = -3$
- $(-2, 5) \implies -2 + 5 = 3$
The pair that satisfies both conditions is $2$ and $-5$.

Step 3: Splitting the Middle Term
We rewrite the middle term $-3x$ as $2x - 5x$:
$x^2 + 2x - 5x - 10 = 0$

Step 4: Grouping and Factoring
Group the terms into two pairs:
$(x^2 + 2x) - (5x + 10) = 0$
Factor out the common terms from each group:
$x(x + 2) - 5(x + 2) = 0$
Now, factor out the common binomial $(x + 2)$:
$(x + 2)(x - 5) = 0$

Step 5: Finding the Roots
According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero.
Case 1: $x + 2 = 0$
$x = -2$
Case 2: $x - 5 = 0$
$x = 5$

Final Answer: The roots of the quadratic equation $x^2 - 3x - 10 = 0$ are $x = -2$ and $x = 5$.