Precalculus Examples

$(x - 3) (x + 2) = 36$

Step 1

Simplify $(x - 3) (x + 2)$ .

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Step 1.1

Expand $(x - 3) (x + 2)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$x (x + 2) - 3 (x + 2) = 36$

Step 1.1.2

Apply the distributive property.

$x \cdot x + x \cdot 2 - 3 (x + 2) = 36$

Step 1.1.3

Apply the distributive property.

$x \cdot x + x \cdot 2 - 3 x - 3 \cdot 2 = 36$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 2 - 3 x - 3 \cdot 2 = 36$

Step 1.2.1.2

Move $2$ to the left of $x$ .

$x^{2} + 2 \cdot x - 3 x - 3 \cdot 2 = 36$

Step 1.2.1.3

Multiply $- 3$ by $2$ .

$x^{2} + 2 x - 3 x - 6 = 36$

Step 1.2.2

Subtract $3 x$ from $2 x$ .

$x^{2} - x - 6 = 36$

Step 2

Subtract $36$ from both sides of the equation.

$x^{2} - x - 6 - 36 = 0$

Step 3

Subtract $36$ from $- 6$ .

$x^{2} - x - 42 = 0$

Step 4

Factor $x^{2} - x - 42$ using the AC method.

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Step 4.1

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 $- 42$ and whose sum is $- 1$ .

$- 7, 6$

Step 4.2

Write the factored form using these integers.

$(x - 7) (x + 6) = 0$

Step 5

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

$x - 7 = 0$

$x + 6 = 0$

Step 6

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 6.2

Add $7$ to both sides of the equation.

$x = 7$

Step 7

Set $x + 6$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 7.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 8

The final solution is all the values that make $(x - 7) (x + 6) = 0$ true.

$x = 7, - 6$