# Solving quadratic equations using factorisation

A quadratic equation is an equation of the second degree; the exponent of one variable is 2. The following are examples of quadratic equations:

$x^2-5x=12$

$a(a-3)-10=0$

$\frac{3b}{b+2}+1=\frac{4}{b+1}$

A quadratic equation has at most two solutions, also referred to as roots. There are some situations, however, in which a quadratic equation has either one solution or no solutions.

One method for solving quadratic equations is factorisation. The standard form of a quadratic equation is ax2 + bx + c = 0 and it is the starting point for solving any equation by factorisation.

It is very important to note that one side of the equation must be equal to zero.

To obtain the two roots we use the fact that if a x b = 0, then a = 0 and/or b = 0. This is called the zero product law.

• Rewrite the equation in the standard form ax2 + bx + c = 0.
• Divide the entire equation by any common factor of the coefficients to obtain a simpler equation of the form ax2 +bx+c = 0, where a, b and c have no common factors.
• Factorise ax2 + bx + c = 0 to be of the form (rx + s) (ux + v) = 0.
• The two solutions are

• Always check the solution by substituting the answer back into the original equation.

Example 1:

Solve for x : x(x – 3 ) = 10

Solution:

Step 1: Rewrite the equation in the form $ax^2+bx+c=0$

Expand the brackets and subtract 10 from both sides of the equation $x^2-3x-10=0$

Step 2: Factorise

(x + 2) (x – 5) = 0

Step 3: Solve for both factors

x + 2 = 0

x = – 2

or

x – 5 = 0

x = 5

The graph shows the roots of the equation x = -2 or x = 5. This graph does not form part of the answer as the question did not ask for a sketch. It is shown here for illustration purposes only.

Step 4: Check the solution by substituting both answers back into the original equation

Step 5: Write the final answer

Therefore x = 2 or x = 5.

Example 2:

Solve the equation: 2x2 – 5x – 12 = 0

Solution:

Step 1: There are no common factors

Step 2: The quadratic equation is already in the standard form ax2 + bx + c = 0

Step 3: Factorise

We must determine the combination of factors of 2 and 12 that will give a middle term coefficient of 5. We find that 2 x 1 and 3 x 4 give a middle term coefficient of 5 so we can factorise the equation as

(2x + 3)(x – 4) = 0

Step 4: Solve for both roots

2x + 3 = 0

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

Or

x – 4 = 0

x = 4

Step 5: Check the solution by substituting both answers back into the original equation

Step 6: Write the final answer

Therefore, $x=-\frac{3}{2}$ or x = 4.