Algebra Examples

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

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

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

Solve for $x$ .

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

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

Move $16$ to the left side of the equation by adding it to both sides.

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

Factor the left side of the equation.

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

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

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

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

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

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

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

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

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

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

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

Factor.

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

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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 $6$ .

$- 2, 8$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

Multiply each term in $- (x - 2) (x + 8) = 0$ by $- 1$

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Multiply each term in $- (x - 2) (x + 8) = 0$ by $- 1$ .

$(- (x - 2) (x + 8)) \cdot - 1 = 0 \cdot - 1$

Simplify $(- (x - 2) (x + 8)) \cdot - 1$ .

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Simplify by multiplying through.

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Apply the distributive property.

$(- x - - 2) (x + 8) \cdot - 1 = 0 \cdot - 1$

Multiply $- 1$ by $- 2$ .

$(- x + 2) (x + 8) \cdot - 1 = 0 \cdot - 1$

Expand $(- x + 2) (x + 8)$ using the FOIL Method.

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Apply the distributive property.

$(- x (x + 8) + 2 (x + 8)) \cdot - 1 = 0 \cdot - 1$

Apply the distributive property.

$(- x \cdot x - x \cdot 8 + 2 (x + 8)) \cdot - 1 = 0 \cdot - 1$

Apply the distributive property.

$(- x \cdot x - x \cdot 8 + 2 x + 2 \cdot 8) \cdot - 1 = 0 \cdot - 1$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ by adding the exponents.

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Move $x$ .

$(- (x \cdot x) - x \cdot 8 + 2 x + 2 \cdot 8) \cdot - 1 = 0 \cdot - 1$

Multiply $x$ by $x$ .

$(- x^{2} - x \cdot 8 + 2 x + 2 \cdot 8) \cdot - 1 = 0 \cdot - 1$

Multiply $8$ by $- 1$ .

$(- x^{2} - 8 x + 2 x + 2 \cdot 8) \cdot - 1 = 0 \cdot - 1$

Multiply $2$ by $8$ .

$(- x^{2} - 8 x + 2 x + 16) \cdot - 1 = 0 \cdot - 1$

Add $- 8 x$ and $2 x$ .

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

Apply the distributive property.

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

Simplify.

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Multiply $- x^{2} \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

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

Multiply $x^{2}$ by $1$ .

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

Multiply $- 1$ by $- 6$ .

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

Multiply $16$ by $- 1$ .

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

Multiply $0$ by $- 1$ .

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

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

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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 $6$ .

$- 2, 8$

Write the factored form using these integers.

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

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

$x - 2 = 0$

$x + 8 = 0$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

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

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

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

$x = 2, - 8$

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