Factorise and solve the quadratic equation x^2 + 6x + 8 = 0

Theory

Problems like this need to be broken down into steps: i) Factorise, ii) solve and iii) check. The aim of this question is to find values of x which satisfy the quadratic equation, i.e. when we substitute our solutions of x into x² + 6x + 8, we get zero. Quadratic factorisation means converting a quadratic equation (an equation with x² in, as provided in the question) into the form: (ax+b)(cx+d) = 0, where a, b, c and d are numbers to be determined. Since 0x0 = 0 we can assume (ax+b) = 0 and (cx+d) = 0 and solve for x in both cases.

Solution

i) Factorise

We want to convert out quadratic equation into (ax+b)(cx+d) = 0. Firstly, multiply out the brackets using the F.O.I.L method – multiply the First terms in each bracket, e.g. ax times cx to give acx², the Outer terms, Inner terms and Last terms, and finally add them all together. This returns acx² + adx + bcx + bd = 0, which we can simplify to ax² + (ad+bc)x + bd = 0.

Now we compare our quadratic equation x² + 6x + 8 = 0 to acx² + (ad+bc)x + bd = 0.

Notice that the coefficient (number in front) of the x² term is 1 in the first equation and ac in the second equation. Hence, ac = 1. We can now set a = 1 and c = 1.

Next, we have (ad + bc) = 6. Using a = 1 and c = 1, this can be simplified to d + b = 6. Similarly, bd = 8.

By trial and error, we can find two numbers that multiply together to make 8 and add together to make 6. These are d = 4 and b = 2.

Substituting a = 1, b = 2, c = 1 and d = 4 into (ax+b)(cx+d) = 0 our factorised quadratic equation is: (x+2)(x+4) = 0.

ii) Solve

Since both brackets in (x+2)(x+4) = 0 are just two numbers multiplied together, both brackets must separately equal zero. Therefore, x+2 = 0 and x+4 = 0, which means x = -2 and x = -4.

iii) Check

Substituting x = -2 and x = -4 into our original equation should return zero:

When x = -2,

(-2)² + 6(-2) + 8 = 4 – 12 + 8 = 0

When x = -4,

(-4)² + 6(-4) + 8 = 16 – 24 + 8 = 0

This is correct, and our solution is: x = -2 and x = -4.

With practise this becomes much easier and steps can be skipped. For example, we would skip straight to comparing x² + 6x + 8 = 0 to acx² + (ad+bc)x + bd = 0, and guessing values for a, b, c and d.