b To find where a function intercepts the

x -

axis, the function can be set equal to zero. The

x -

values that satisfy the equation are the roots of the function. Thus, to find the roots of the given quadratic equation we need to solve it for

x .

To do this we can use factoring and the Zero Product Property.

Here we have a quadratic equation of the form

0 = a x^{2} + b x + c,

where

∣ a ∣ \neq = 1

and there are no common factors. To factor this equation we will rewrite the middle term,

b x,

as two terms. The coefficients of these two terms will be factors of

a c

whose sum must be

b .

0 = 6 x^{2} - 23 x + 20 ⇕ 0 = 6 x^{2} + (- 23) x + 20

We have that

a = 6,

b = - 23,

and

c = 20 .

There are now three steps we need to follow in order to rewrite the above equation.

Find $a c .$ Since we have that $a = 6$ and $c = 20,$ the value of $a c$ is $6 \times 20 = 120 .$
Find factors of $a c .$ Since $a c = 120,$ which is positive, we need factors of $a c$ to have the same sign — both positive or both negative — in order for the product to be positive. Since $b = - 23,$ which is negative, those factors will need to be negative so that their sum is negative.

\underline{1^{st} Factor} - 1 - 2 - 3 - 4 - 5 - 6 - 8 - 10 \underline{2^{nd} Factor} - 120 - 60 - 40 - 30 - 24 - 20 - 15 - 12 \underline{Sum} - 1 + (- 120) - 2 + (- 60) - 3 + (- 40) - 4 + (- 30) - 5 + (- 24) - 6 + (- 20) - 8 + (- 15) - 10 + (- 12) \underline{Result} - 121 - 62 - 43 - 34 - 29 - 26 - 23 - 22

Rewrite $b x$ as two terms. Now that we know which factors are the ones to be used, we can rewrite $b x$ as two terms.
$0 = 6 x^{2} + (- 23) x + 20 ⇕ 0 = 6 x^{2} - 8 x - 15 x + 20$

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