Finding the zeros of a quadratic by factoring

Question

Solve for x when f(x)=0 where f(x)=6x²+7x+2

Answer 1

You can solve a quadratic equation by factoring (if possible)

or using the quadratic formula x = (–b ±√(b² – 4ac))/2a

Using the Quadratic Formula

Using the quadratic formula for 6x² + 7x + 2 = 0 you would calculate

x = (–7 ±√(7² – 4(6)(2)))/2(6) = (–7 ±√(49 – 48))/12 = (–7 ±1)/12

So you two solutions would be

1. x = (–7 +1)/12 = –6/12 =
2. x = (–7 – 1)/12 = –8/12 =

You can also factor 6x² + 7x + 2 if the question requires it—as it does here apparently.

Solving by Factoring

The 1^st step is to find a pair of integers m and n that they

multiply to give m × n = a × c = 6 × 2 = 12

and add to give m + n = b = 7

the numbers m = 3 and n = 4 work

3 × 4 = 12

3 + 4 = 7

The 2^nd step is to replace the middle term of the quadratic, 7x, with 3x + 4x

6x² + 7x + 2 = (6x² + 3x) + (4x + 2) =

The 3^rd step is to factor each pair of terms

(6x² + 3x) + (4x + 2) = 3x(2x + 1) + 2(2x + 1)

The last step is to recognize that the common factor (2x + 1) in each term means that

you can rewrite the equation in factored form

3x(2x + 1) + 2(2x + 1) = (3x + 2)(2x + 1)

(3x + 2)(2x + 1) = 0 means that your two solutions come from setting each factor to zero

3x + 2 = 0 gives you one solution

2x + 1 = 0 gives you the other solution

These, of course, will be the same two solutions you get from the quadratic formula.

Sometimes factoring can be the quicker route, but in this case you get to the answer faster using the quadratic formula.

However, even if you are required to factor, you can use the quadratic formula to check your answer.