Sometimes, a quadratic expression looks complicated, and it would help out if it could be simplified. This can be through using decomposition. Another way though could be to find the greatest common factor of the a, b and c terms and divide the entire expression by this value. Take the expression in (eq. 1) for example:

Eq. 1: \(y=2x^2+20x+48\)

To find the greatest common factor of this expression, we need to find the factors for each of the numbers 2, 20 and 48. To do this, we need to separately find the factors of each value (Table 1).

Table 1: The factors of the numbers 2, 20 and 48. The greatest common factor is 2, highlighted in bold.

With the greatest common factor of 2 identified, the entire expression can be divided by 2 and place inside a bracket with 2 on the outside (Eq. 2). This indicates that the entire expression is multiplied by 2, and thus gives the same value as the expression did before dividing by the greatest common factor.

Eq. 2: \(y=2(x^2+10x+24)\)

You can follow the process to factorize this expression to turn the expression inside the bracket into the factored form (Eq. 3). The factored form turns the single bracket into two brackets by themselves, not into two brackets inside another bracket, as there are three terms that are multiplied together, 2, and each of the expressions of the factored form inside the brackets.

Eq. 3: \(y=2(x+6)(x+4)\)

To verify that the factored form following finding the greatest common factor, we can apply the FOIL method to the expression, only this time, because we have an extra term (2), we have to multiply all of our FOIL expressions to this term too (Eq. 4, Eq. 5, Eq. 6 and Eq. 7).

Eq. 4: \(2\times x\times x=2x^2\)

Eq. 5: \(2\times x\times 4=8x\)

Eq. 6: \(2\times 6\times x=12x\)

Eq. 7: \(2\times 6\times 4=48\)

These values can be combined to give the original (Eq. 1).