Calculus Examples

$f (x) = 6 x - 16$ , $f (x) = - x^{2}$

Step 1

Substitute $- x^{2}$ for $f (x)$ .

$- x^{2} = 6 x - 16$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $6 x$ from both sides of the equation.

$- x^{2} - 6 x = - 16$

Step 2.2

Add $16$ to both sides of the equation.

$- x^{2} - 6 x + 16 = 0$

Step 2.3

Factor the left side of the equation.

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Step 2.3.1

Factor $- 1$ out of $- x^{2} - 6 x + 16$ .

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Step 2.3.1.1

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) - 6 x + 16 = 0$

Step 2.3.1.2

Factor $- 1$ out of $- 6 x$ .

$- (x^{2}) - (6 x) + 16 = 0$

Step 2.3.1.3

Rewrite $16$ as $- 1 (- 16)$ .

$- (x^{2}) - (6 x) - 1 \cdot - 16 = 0$

Step 2.3.1.4

Factor $- 1$ out of $- (x^{2}) - (6 x)$ .

$- (x^{2} + 6 x) - 1 \cdot - 16 = 0$

Step 2.3.1.5

Factor $- 1$ out of $- (x^{2} + 6 x) - 1 (- 16)$ .

$- (x^{2} + 6 x - 16) = 0$

Step 2.3.2

Factor.

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Step 2.3.2.1

Factor $x^{2} + 6 x - 16$ using the AC method.

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Step 2.3.2.1.1

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 $- 16$ and whose sum is $6$ .

$- 2, 8$

Step 2.3.2.1.2

Write the factored form using these integers.

$- ((x - 2) (x + 8)) = 0$

Step 2.3.2.2

Remove unnecessary parentheses.

$- (x - 2) (x + 8) = 0$

Step 2.4

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

$x - 2 = 0$

$x + 8 = 0$

Step 2.5

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.6

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

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Step 2.6.1

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

$x + 8 = 0$

Step 2.6.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 2.7

The final solution is all the values that make $- (x - 2) (x + 8) = 0$ true.

$x = 2, - 8$

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