To factor the quadratic expression, we will start by identifying the values of

a,

b,

and

c .

y^{2} - 7 y - 30 \Leftrightarrow 1 y^{2} + (- 7) y + (- 30)

For our expression, we have that

a = 1,

b = - 7,

and

c = - 30 .

To factor a quadratic expression with leading coefficient

a = 1,

we need to find two factors of

c = - 30

whose sum is

b = - 7 .

Since

- 30

is a negative number, we will only consider factors with opposite signs — one positive and one negative — so that their product is negative.

Factor Pair	Product	Sum
$1$ and $- 30$	$- 30^{1 \times (- 30)}$	$- 29 1 + (- 30)$
$- 1$ and $30$	$- 30^{- 1 \times 30}$	$29^{- 1 + 30}$
$2$ and $- 15$	$- 30^{2 \times (- 15)}$	$- 13^{2 + (- 15)}$
$- 2$ and $15$	$- 30^{- 2 \times 15}$	$13^{- 2 + 15}$
$3$ and $- 10$	$- 30^{3 \times (- 10)}$	$- 7^{3 + (- 10)}$
$- 3$ and $10$	$- 30^{- 3 \times 10}$	$7^{- 3 + 10}$
$5$ and $- 6$	$- 30^{5 \times (- 6)}$	$- 1^{5 + (- 6)}$
$- 5$ and $6$	$- 30^{- 5 \times 6}$	$1^{- 5 + 6}$

The integers whose product is

- 30

and whose sum is

- 7

are

3

and

- 10 .

y^{2} - 7 y - 30 \Leftrightarrow (y + 3) (y - 10)

Let's use a graphing calculator to confirm our answer. To do so, we will graph the related functions in the same coordinate plane. Note that in the calculator we will use the variable

x

instead of

y .

We see that only one graph appears. This means that both graphs coincide. Therefore, the expression has been factored correctly.

Exercise

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