Method of Solving a Quadratic Equation - Factorisation Method

The factorisation method expresses a quadratic polynomial as a product of two linear factors.

If \[ ax^{2} + bx + c = (px + q)(rx + s), \]

then the roots are obtained by solving:

\[ px + q = 0 \] and

\[ rx + s = 0. \]

If product of two numbers is zero then at least one of them is zero.

\[ p \times q = 0 \;\Rightarrow\; p = 0 \;\text{ or }\; q = 0 \]

Applied:

\[ x^{2} - 4x - 5 = (x - 5)(x + 1) = 0 \]

\[ \therefore\; x - 5 = 0 \;\text{ or }\; x + 1 = 0 \]

\[ \therefore\; x = 5 \;\text{ or }\; x = -1 \]

Solve: \[ x^{2} - 4x - 5 = 0 \]

Solution: Find two numbers whose product is \[ -5 \] and sum is \[ -4. \]

The numbers are \[ -5 \] and \[ 1. \]

\[ x^{2} - 5x + x - 5 = 0 \]

\[ x(x - 5) + 1(x - 5) = 0 \]

\[ (x - 5)(x + 1) = 0 \]

Using the zero-product rule:

\[ x - 5 = 0 \quad \text{or} \quad x + 1 = 0 \]

\[ \boxed{x = 5 \text{ or } x = -1} \]

Verification:

For \[ x = 5: \]

\[ 25 - 20 - 5 = 0 \]

For \[ x = -1: \]

\[ 1 + 4 - 5 = 0 \]

Hence, both roots are correct.

Solve: \[ 6\sqrt{3}x^{2} + 7x = \sqrt{3} \]

Step 1: Express the equation in standard form

\[ 6\sqrt{3}x^{2} + 7x = \sqrt{3} \]

\[ \therefore\; 6\sqrt{3}x^{2} + 7x - \sqrt{3} = 0 \ldots \]

Step 2: Split the middle term

\[ 6\sqrt{3} \times (-\sqrt{3}) = -18 \]

Factors of \[ -18 \] : \[ 9 \] and \[ -2 \]

\[ 9 = 3\sqrt{3} \times \sqrt{3} \]

\[ \therefore\; 6\sqrt{3}x^{2} + 9x - 2x - \sqrt{3} = 0 \]

Step 3: Factorise by grouping

\[ \therefore\; 3\sqrt{3}x(2x + \sqrt{3}) - 1(2x + \sqrt{3}) = 0 \]

\[ \therefore\; (2x + \sqrt{3})(3\sqrt{3}x - 1) = 0 \]

Step 4: Find the roots

\[ \therefore\; 2x + \sqrt{3} = 0 \quad \text{or} \quad 3\sqrt{3}x - 1 = 0 \]

\[ \therefore\; 2x = -\sqrt{3} \quad \text{or} \quad 3\sqrt{3}x = 1 \]

\[ \therefore\; x = -\frac{\sqrt{3}}{2} \quad \text{or} \quad x = \frac{1}{3\sqrt{3}} \]

Final Answer:

\[ \therefore\; -\frac{\sqrt{3}}{2} \] and \[ \frac{1}{3\sqrt{3}} \] are the roots of the given quadratic equation.

In the factorisation method, the quadratic expression is written as a product of two linear factors

1. Clear fractions and brackets, if any.

2. Transpose all terms to one side to get the standard form
   ax² + bx + c = 0
3. Factorise the quadratic expression into two linear factors.

4. Put each factor equal to zero (using the zero product rule).

5. Solve the resulting linear equations to obtain the roots.