Precalculus Examples

$x^{2} = 3 x + 40$

Step 1

Subtract $3 x$ from both sides of the equation.

$x^{2} - 3 x = 40$

Step 2

Subtract $40$ from both sides of the equation.

$x^{2} - 3 x - 40 = 0$

Step 3

Factor $x^{2} - 3 x - 40$ using the AC method.

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Step 3.1

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 $- 40$ and whose sum is $- 3$ .

$- 8, 5$

Step 3.2

Write the factored form using these integers.

$(x - 8) (x + 5) = 0$

Step 4

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

$x - 8 = 0$

$x + 5 = 0$

Step 5

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 5.2

Add $8$ to both sides of the equation.

$x = 8$

Step 6

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

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Step 6.1

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

$x + 5 = 0$

Step 6.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 7

The final solution is all the values that make $(x - 8) (x + 5) = 0$ true.

$x = 8, - 5$