Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- \frac{1}{2} \cdot (x (x - 8)) = 0$ .

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

Step 1.2.2

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

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

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

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Step 1.2.3.1.1.1

Apply the distributive property.

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

Step 1.2.3.1.1.2

Simplify the expression.

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Step 1.2.3.1.1.2.1

Multiply $x$ by $x$ .

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

Step 1.2.3.1.1.2.2

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

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

Step 1.2.3.1.1.3

Apply the distributive property.

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

Step 1.2.3.1.1.4

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

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

Step 1.2.3.1.1.5

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.5.1

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

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

Step 1.2.3.1.1.5.2

Factor $2$ out of $- 8 x$ .

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

Step 1.2.3.1.1.5.3

Cancel the common factor.

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

Step 1.2.3.1.1.5.4

Rewrite the expression.

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

Step 1.2.3.1.1.6

Multiply $- 4$ by $- 1$ .

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

Step 1.2.3.1.1.7

Apply the distributive property.

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

Step 1.2.3.1.1.8

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.8.1

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

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

Step 1.2.3.1.1.8.2

Factor $2$ out of $- 2$ .

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

Step 1.2.3.1.1.8.3

Cancel the common factor.

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

Step 1.2.3.1.1.8.4

Rewrite the expression.

$- - x^{2} - 2 (4 x) = - 2 \cdot 0$

Step 1.2.3.1.1.9

Multiply.

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Step 1.2.3.1.1.9.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.3.1.1.9.2

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

$x^{2} - 2 (4 x) = - 2 \cdot 0$

Step 1.2.3.1.1.9.3

Multiply $4$ by $- 2$ .

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $- 2$ by $0$ .

$x^{2} - 8 x = 0$

Step 1.2.4

Factor $x$ out of $x^{2} - 8 x$ .

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Step 1.2.4.1

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

$x \cdot x - 8 x = 0$

Step 1.2.4.2

Factor $x$ out of $- 8 x$ .

$x \cdot x + x \cdot - 8 = 0$

Step 1.2.4.3

Factor $x$ out of $x \cdot x + x \cdot - 8$ .

$x (x - 8) = 0$

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 $0$ .

$x = 0$

$x - 8 = 0$

Step 1.2.6

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.7

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

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Step 1.2.7.1

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

$x - 8 = 0$

Step 1.2.7.2

Add $8$ to both sides of the equation.

$x = 8$

Step 1.2.8

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

$x = 0, 8$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (8, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Remove parentheses.

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

Step 2.2.4

Simplify $- \frac{1}{2} \cdot ((0) ((0) - 8))$ .

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Step 2.2.4.1

Cancel the common factor of $2$ .

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Step 2.2.4.1.1

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

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

Step 2.2.4.1.2

Factor $2$ out of $(0) ((0) - 8)$ .

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

Step 2.2.4.1.3

Cancel the common factor.

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

Step 2.2.4.1.4

Rewrite the expression.

$y = - 1 \cdot ((0) ((0) - 8))$

Step 2.2.4.2

Multiply by zero.

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Step 2.2.4.2.1

Multiply $0$ by $(0) - 8$ .

$y = - 1 \cdot 0$

Step 2.2.4.2.2

Multiply $- 1$ by $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (8, 0)$

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

Step 4