Precalculus Examples

$x^{2} + 8 x = 65$

Step 1

Subtract $65$ from both sides of the equation.

$x^{2} + 8 x - 65 = 0$

Step 2

Factor $x^{2} + 8 x - 65$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 65$ and whose sum is $8$ .

$- 5, 13$

Step 2.2

Write the factored form using these integers.

$(x - 5) (x + 13) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 5 = 0$

$x + 13 = 0$

Step 4

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 4.2

Add $5$ to both sides of the equation.

$x = 5$

Step 5

Set $x + 13$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x + 13$ equal to $0$ .

$x + 13 = 0$

Step 5.2

Subtract $13$ from both sides of the equation.

$x = - 13$

Step 6

The final solution is all the values that make $(x - 5) (x + 13) = 0$ true.

$x = 5, - 13$