Algebra Examples

$y = \frac{1}{4} (8 x - x^{2})$

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 $x$ .

$0 = \frac{1}{4} \cdot (8 x - x^{2})$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{1}{4} \cdot (8 x - x^{2}) = 0$ .

$\frac{1}{4} \cdot (8 x - x^{2}) = 0$

Step 1.2.2

Multiply both sides of the equation by $4$ .

$4 (\frac{1}{4} \cdot (8 x - x^{2})) = 4 \cdot 0$

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 $4 (\frac{1}{4} \cdot (8 x - x^{2}))$ .

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Step 1.2.3.1.1.1

Apply the distributive property.

$4 (\frac{1}{4} (8 x) + \frac{1}{4} (- x^{2})) = 4 \cdot 0$

Step 1.2.3.1.1.2

Cancel the common factor of $4$ .

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Step 1.2.3.1.1.2.1

Factor $4$ out of $8 x$ .

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

Step 1.2.3.1.1.2.2

Cancel the common factor.

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

Step 1.2.3.1.1.2.3

Rewrite the expression.

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

Step 1.2.3.1.1.3

Combine $\frac{1}{4}$ and $x^{2}$ .

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

Step 1.2.3.1.1.4

Apply the distributive property.

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

Step 1.2.3.1.1.5

Multiply $2$ by $4$ .

$8 x + 4 (- \frac{x^{2}}{4}) = 4 \cdot 0$

Step 1.2.3.1.1.6

Cancel the common factor of $4$ .

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Step 1.2.3.1.1.6.1

Move the leading negative in $- \frac{x^{2}}{4}$ into the numerator.

$8 x + 4 \frac{- x^{2}}{4} = 4 \cdot 0$

Step 1.2.3.1.1.6.2

Cancel the common factor.

$8 x + 4 \frac{- x^{2}}{4} = 4 \cdot 0$

Step 1.2.3.1.1.6.3

Rewrite the expression.

$8 x - x^{2} = 4 \cdot 0$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $4$ by $0$ .

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

Step 1.2.4

Factor the left side of the equation.

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Step 1.2.4.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

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

Step 1.2.4.2

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

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Step 1.2.4.2.1

Factor $u$ out of $8 u$ .

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

Step 1.2.4.2.2

Factor $u$ out of $- u^{2}$ .

$u \cdot 8 + u (- u) = 0$

Step 1.2.4.2.3

Factor $u$ out of $u \cdot 8 + u (- u)$ .

$u (8 - u) = 0$

Step 1.2.4.3

Replace all occurrences of $u$ with $x$ .

$x (8 - x) = 0$

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 $0$ .

$x = 0$

$8 - x = 0$

Step 1.2.6

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.7

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

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Step 1.2.7.1

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

$8 - x = 0$

Step 1.2.7.2

Solve $8 - x = 0$ for $x$ .

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Step 1.2.7.2.1

Subtract $8$ from both sides of the equation.

$- x = - 8$

Step 1.2.7.2.2

Divide each term in $- x = - 8$ by $- 1$ and simplify.

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Step 1.2.7.2.2.1

Divide each term in $- x = - 8$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 8}{- 1}$

Step 1.2.7.2.2.2

Simplify the left side.

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Step 1.2.7.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 8}{- 1}$

Step 1.2.7.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 8}{- 1}$

Step 1.2.7.2.2.3

Simplify the right side.

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Step 1.2.7.2.2.3.1

Divide $- 8$ by $- 1$ .

$x = 8$

Step 1.2.8

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

$y = \frac{1}{4} \cdot (8 (0) - {(0)}^{2})$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{1}{4} \cdot (8 (0) - 0^{2})$

Step 2.2.2

Remove parentheses.

$y = \frac{1}{4} \cdot (8 (0) - {(0)}^{2})$

Step 2.2.3

Simplify $\frac{1}{4} \cdot (8 (0) - {(0)}^{2})$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $8$ by $0$ .

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

Step 2.2.3.1.2

Raising $0$ to any positive power yields $0$ .

$y = \frac{1}{4} \cdot (0 - 0)$

Step 2.2.3.1.3

Multiply $- 1$ by $0$ .

$y = \frac{1}{4} \cdot (0 + 0)$

Step 2.2.3.2

Simplify the expression.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = \frac{1}{4} \cdot 0$

Step 2.2.3.2.2

Multiply $\frac{1}{4}$ 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