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.
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