Algebra Examples

$f (x) = \frac{1}{2} (x - 2) (x + 6)$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{1}{2} \cdot ((x - 2) (x + 6)) = 0$ .

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

Step 1.2.2

Multiply both sides of the equation by $2$ .

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

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Simplify $2 (\frac{1}{2} \cdot ((x - 2) (x + 6)))$ .

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Step 1.2.3.1.1.1

Expand $(x - 2) (x + 6)$ using the FOIL Method.

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Step 1.2.3.1.1.1.1

Apply the distributive property.

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

Step 1.2.3.1.1.1.2

Apply the distributive property.

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

Step 1.2.3.1.1.1.3

Apply the distributive property.

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

Step 1.2.3.1.1.2

Simplify and combine like terms.

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Step 1.2.3.1.1.2.1

Simplify each term.

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Step 1.2.3.1.1.2.1.1

Multiply $x$ by $x$ .

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

Step 1.2.3.1.1.2.1.2

Move $6$ to the left of $x$ .

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

Step 1.2.3.1.1.2.1.3

Multiply $- 2$ by $6$ .

$2 (\frac{1}{2} \cdot (x^{2} + 6 x - 2 x - 12)) = 2 \cdot 0$

Step 1.2.3.1.1.2.2

Subtract $2 x$ from $6 x$ .

$2 (\frac{1}{2} \cdot (x^{2} + 4 x - 12)) = 2 \cdot 0$

Step 1.2.3.1.1.3

Apply the distributive property.

$2 (\frac{1}{2} x^{2} + \frac{1}{2} (4 x) + \frac{1}{2} \cdot - 12) = 2 \cdot 0$

Step 1.2.3.1.1.4

Simplify.

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Step 1.2.3.1.1.4.1

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

$2 (\frac{x^{2}}{2} + \frac{1}{2} (4 x) + \frac{1}{2} \cdot - 12) = 2 \cdot 0$

Step 1.2.3.1.1.4.2

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.4.2.1

Factor $2$ out of $4 x$ .

$2 (\frac{x^{2}}{2} + \frac{1}{2} (2 (2 x)) + \frac{1}{2} \cdot - 12) = 2 \cdot 0$

Step 1.2.3.1.1.4.2.2

Cancel the common factor.

$2 (\frac{x^{2}}{2} + \frac{1}{2} (2 (2 x)) + \frac{1}{2} \cdot - 12) = 2 \cdot 0$

Step 1.2.3.1.1.4.2.3

Rewrite the expression.

$2 (\frac{x^{2}}{2} + 2 x + \frac{1}{2} \cdot - 12) = 2 \cdot 0$

Step 1.2.3.1.1.4.3

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.4.3.1

Factor $2$ out of $- 12$ .

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

Step 1.2.3.1.1.4.3.2

Cancel the common factor.

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

Step 1.2.3.1.1.4.3.3

Rewrite the expression.

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

Step 1.2.3.1.1.5

Apply the distributive property.

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

Step 1.2.3.1.1.6

Simplify.

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Step 1.2.3.1.1.6.1

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.6.1.1

Cancel the common factor.

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

Step 1.2.3.1.1.6.1.2

Rewrite the expression.

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

Step 1.2.3.1.1.6.2

Multiply $2$ by $2$ .

$x^{2} + 4 x + 2 \cdot - 6 = 2 \cdot 0$

Step 1.2.3.1.1.6.3

Multiply $2$ by $- 6$ .

$x^{2} + 4 x - 12 = 2 \cdot 0$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $2$ by $0$ .

$x^{2} + 4 x - 12 = 0$

Step 1.2.4

Factor $x^{2} + 4 x - 12$ using the AC method.

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Step 1.2.4.1

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 $- 12$ and whose sum is $4$ .

$- 2, 6$

Step 1.2.4.2

Write the factored form using these integers.

$(x - 2) (x + 6) = 0$

Step 1.2.5

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

$x - 2 = 0$

$x + 6 = 0$

Step 1.2.6

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

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Step 1.2.6.1

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

$x - 2 = 0$

Step 1.2.6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.7

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

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Step 1.2.7.1

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

$x + 6 = 0$

Step 1.2.7.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 1.2.8

The final solution is all the values that make $(x - 2) (x + 6) = 0$ true.

$x = 2, - 6$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(2, 0), (- 6, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{1}{2} \cdot (((0) - 2) ((0) + 6))$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{1}{2} \cdot ((0 - 2) ((0) + 6))$

Step 2.2.2

Remove parentheses.

$y = \frac{1}{2} \cdot ((0 - 2) (0 + 6))$

Step 2.2.3

Remove parentheses.

$y = \frac{1}{2} \cdot ((0 - 2) (0 + 6))$

Step 2.2.4

Remove parentheses.

$y = \frac{1}{2} \cdot (((0) - 2) ((0) + 6))$

Step 2.2.5

Simplify $\frac{1}{2} \cdot (((0) - 2) ((0) + 6))$ .

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Step 2.2.5.1

Subtract $2$ from $0$ .

$y = \frac{1}{2} \cdot (- 2 ((0) + 6))$

Step 2.2.5.2

Cancel the common factor of $2$ .

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Step 2.2.5.2.1

Factor $2$ out of $- 2 ((0) + 6)$ .

$y = \frac{1}{2} \cdot (2 (- ((0) + 6)))$

Step 2.2.5.2.2

Cancel the common factor.

$y = \frac{1}{2} \cdot (2 (- ((0) + 6)))$

Step 2.2.5.2.3

Rewrite the expression.

$y = - ((0) + 6)$

Step 2.2.5.3

Simplify the expression.

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Step 2.2.5.3.1

Add $0$ and $6$ .

$y = - 1 \cdot 6$

Step 2.2.5.3.2

Multiply $- 1$ by $6$ .

$y = - 6$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 6)$

Step 3

List the intersections.

x-intercept(s): $(2, 0), (- 6, 0)$

y-intercept(s): $(0, - 6)$

Step 4