A quadratic equation is an equation where the highest power of the variable is \( x^2 \). Solving a quadratic equation means finding the value or values of \( x \) that make the equation equal to zero.

 

This topic focuses on solving equations of the form
\( x^2 + bx + c = 0 \) and \( ax^2 + bx + c = 0 \)
by factorisation.

 

 

Understanding the Zero Product Rule

Factorisation relies on the zero product rule, which states:

 

If
\( ab = 0 \)
then
\( a = 0 \) or \( b = 0 \)

 

This rule allows quadratic equations written as a product of two brackets to be solved.

 

 

Solving Equations of the Form x² + bx + c = 0

First factorise the quadratic, then set each bracket equal to zero.

 

For example:

$$
x^2 + 7x + 10 = 0
$$

 

Factorise:

$$
(x + 5)(x + 2) = 0
$$

 

Set each bracket equal to zero:

$$
x + 5 = 0
$$

$$
x + 2 = 0
$$

 

Solve:

$$
x = -5
$$

$$
x = -2
$$

So the solutions are \( x = -5 \) and \( x = -2 \).

 

Another example:

$$
x^2 - x - 6 = 0
$$

 

Factorise:

$$
(x - 3)(x + 2) = 0
$$

 

So:

$$
x = 3
$$

$$
x = -2
$$

 

 

Solving Equations of the Form ax² + bx + c = 0

When the coefficient of \( x^2 \) is not 1, the equation must still be factorised first.

 

For example:

$$
2x^2 + 7x + 3 = 0
$$

 

Factorise:

$$
(2x + 1)(x + 3) = 0
$$

 

Set each bracket equal to zero:

$$
2x + 1 = 0
$$

$$
x + 3 = 0
$$

 

Solve:

$$
x = -\frac{1}{2}
$$

$$
x = -3
$$

 

Another example:

$$
3x^2 - 5x - 2 = 0
$$

 

Factorise:

$$
(3x + 1)(x - 2) = 0
$$

 

So:

$$
x = -\frac{1}{3}
$$

$$
x = 2
$$

 

 

Checking Solutions

It is good practice to check solutions by substituting them back into the original equation to confirm that they satisfy it.

 

 

Key Points to Remember

Always rearrange the equation so it equals zero.
Factorise the quadratic fully before solving.
Use the zero product rule to find solutions.
Quadratic equations can have two, one or no real solutions.

 

Solving quadratic equations by factorisation is a core algebra skill and forms the basis for more advanced solving methods later on.