Algebra Examples

$f (x) = - \frac{1}{2} (x - 2) (x + 8)$

Step 1

Set $- \frac{1}{2} \cdot ((x - 2) (x + 8))$ equal to $0$ .

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

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides of the equation by $- 2$ .

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

Step 2.2

Simplify both sides of the equation.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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Step 2.2.1.1.1.1

Apply the distributive property.

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

Step 2.2.1.1.1.2

Apply the distributive property.

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

Step 2.2.1.1.1.3

Apply the distributive property.

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

Step 2.2.1.1.2

Simplify and combine like terms.

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Step 2.2.1.1.2.1

Simplify each term.

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Step 2.2.1.1.2.1.1

Multiply $x$ by $x$ .

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

Step 2.2.1.1.2.1.2

Move $8$ to the left of $x$ .

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

Step 2.2.1.1.2.1.3

Multiply $- 2$ by $8$ .

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

Step 2.2.1.1.2.2

Subtract $2 x$ from $8 x$ .

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

Step 2.2.1.1.3

Apply the distributive property.

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

Step 2.2.1.1.4

Simplify.

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Step 2.2.1.1.4.1

Combine $x^{2}$ and $\frac{1}{2}$ .

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

Step 2.2.1.1.4.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 2.2.1.1.4.2.2

Factor $2$ out of $6 x$ .

$- 2 (- \frac{x^{2}}{2} + \frac{- 1}{2} (2 (3 x)) - \frac{1}{2} \cdot - 16) = - 2 \cdot 0$

Step 2.2.1.1.4.2.3

Cancel the common factor.

$- 2 (- \frac{x^{2}}{2} + \frac{- 1}{2} (2 (3 x)) - \frac{1}{2} \cdot - 16) = - 2 \cdot 0$

Step 2.2.1.1.4.2.4

Rewrite the expression.

$- 2 (- \frac{x^{2}}{2} - 1 (3 x) - \frac{1}{2} \cdot - 16) = - 2 \cdot 0$

Step 2.2.1.1.4.3

Multiply $3$ by $- 1$ .

$- 2 (- \frac{x^{2}}{2} - 3 x - \frac{1}{2} \cdot - 16) = - 2 \cdot 0$

Step 2.2.1.1.4.4

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.4.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$- 2 (- \frac{x^{2}}{2} - 3 x + \frac{- 1}{2} \cdot - 16) = - 2 \cdot 0$

Step 2.2.1.1.4.4.2

Factor $2$ out of $- 16$ .

$- 2 (- \frac{x^{2}}{2} - 3 x + \frac{- 1}{2} \cdot (2 (- 8))) = - 2 \cdot 0$

Step 2.2.1.1.4.4.3

Cancel the common factor.

$- 2 (- \frac{x^{2}}{2} - 3 x + \frac{- 1}{2} \cdot (2 \cdot - 8)) = - 2 \cdot 0$

Step 2.2.1.1.4.4.4

Rewrite the expression.

$- 2 (- \frac{x^{2}}{2} - 3 x - 1 \cdot - 8) = - 2 \cdot 0$

Step 2.2.1.1.4.5

Multiply $- 1$ by $- 8$ .

$- 2 (- \frac{x^{2}}{2} - 3 x + 8) = - 2 \cdot 0$

Step 2.2.1.1.5

Apply the distributive property.

$- 2 (- \frac{x^{2}}{2}) - 2 (- 3 x) - 2 \cdot 8 = - 2 \cdot 0$

Step 2.2.1.1.6

Simplify.

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Step 2.2.1.1.6.1

Cancel the common factor of $2$ .

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Step 2.2.1.1.6.1.1

Move the leading negative in $- \frac{x^{2}}{2}$ into the numerator.

$- 2 \frac{- x^{2}}{2} - 2 (- 3 x) - 2 \cdot 8 = - 2 \cdot 0$

Step 2.2.1.1.6.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- x^{2}}{2} - 2 (- 3 x) - 2 \cdot 8 = - 2 \cdot 0$

Step 2.2.1.1.6.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- x^{2}}{2} - 2 (- 3 x) - 2 \cdot 8 = - 2 \cdot 0$

Step 2.2.1.1.6.1.4

Rewrite the expression.

$- - x^{2} - 2 (- 3 x) - 2 \cdot 8 = - 2 \cdot 0$

Step 2.2.1.1.6.2

Multiply $- 1$ by $- 1$ .

$1 x^{2} - 2 (- 3 x) - 2 \cdot 8 = - 2 \cdot 0$

Step 2.2.1.1.6.3

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

$x^{2} - 2 (- 3 x) - 2 \cdot 8 = - 2 \cdot 0$

Step 2.2.1.1.6.4

Multiply $- 3$ by $- 2$ .

$x^{2} + 6 x - 2 \cdot 8 = - 2 \cdot 0$

Step 2.2.1.1.6.5

Multiply $- 2$ by $8$ .

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

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $- 2$ by $0$ .

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

Step 2.3

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

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Step 2.3.1

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

$- 2, 8$

Step 2.3.2

Write the factored form using these integers.

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

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.

$x = 2, - 8$

Step 3