Quadratic equations - Factorization method

We have agreed that there are four methods of solving quadratic equations. The first one is the method of factorization.

This method tends to split quadratic equation into two factors such that when they are multiplied together, they will reveal the same quadratic equation.

Example1

If x² + 6x + 5 = 0, find the possible values of x using factorization method.

Solution.

x² + 6x + 5 = 0

Step1

Find two numbers which, when multiplied will give the constant term (ie 5) and when added (or subtracted) will give the coefficient of x (ie 6)

Factors: factors of 5 are “5 and 1”

Sum: 5+1=6.

Thus, “5 and 1” satisfy the equation.

Step2

Replace 6x with 5x and 1x in the equation. (Note that x is the same as 1x)

We then have x² + 5x + x + 5 = 0 (by splitting 6x into 5x and x)

Step3

Group the first two terms together and the last two terms together as follows

(x² + 5x) + (x + 5) = 0

By bringing out common terms, this becomes

x(x + 5) + 1(x + 5) = 0

Since the two brackets are the same, we take one of them and multiply by the coefficients of the brackets as follows

(x + 5)(x + 1) = 0

If two things are multiplied and the result is zero, it means that either one of them is equal to zero or both are equal to zero

Therefore,

Either x + 5 = 0, which gives x = 0 – 5

And therefore x = –5

Or x + 1 = 0, which gives x = 0 – 1

And therefore x = –1

The values of x that satisfy the equation

x² + 6x + 5 = 0 are x = –5 or -1
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