Algebra Examples

$f (x) = \frac{- 4 x^{2} + 4 x}{5 x^{2} - 25 x + 20}$

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 $x$ .

$0 = \frac{- 4 x^{2} + 4 x}{5 x^{2} - 25 x + 20}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

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

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Factor $- 4 x$ out of $- 4 x^{2} + 4 x$ .

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Step 1.2.2.1.1

Factor $- 4 x$ out of $- 4 x^{2}$ .

$- 4 x \cdot x + 4 x = 0$

Step 1.2.2.1.2

Factor $- 4 x$ out of $4 x$ .

$- 4 x \cdot x - 4 x \cdot - 1 = 0$

Step 1.2.2.1.3

Factor $- 4 x$ out of $- 4 x (x) - 4 x (- 1)$ .

$- 4 x (x - 1) = 0$

Step 1.2.2.2

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

$x = 0$

$x - 1 = 0$

Step 1.2.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.2.4

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 1.2.2.4.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 1.2.2.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.2.5

The final solution is all the values that make $- 4 x (x - 1) = 0$ true.

$x = 0, 1$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{- 4 x^{2} + 4 x}{5 x^{2} - 25 x + 20}$ true.

$x = 0$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

$y = \frac{- 4 {(0)}^{2} + 4 (0)}{5 {(0)}^{2} - 25 \cdot 0 + 20}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{- 4 \cdot 0^{2} + 4 (0)}{5 {(0)}^{2} - 25 \cdot 0 + 20}$

Step 2.2.2

Remove parentheses.

$y = \frac{- 4 \cdot 0^{2} + 4 (0)}{5 \cdot 0^{2} - 25 \cdot 0 + 20}$

Step 2.2.3

Remove parentheses.

$y = \frac{- 4 {(0)}^{2} + 4 (0)}{5 {(0)}^{2} - 25 \cdot 0 + 20}$

Step 2.2.4

Simplify $\frac{- 4 {(0)}^{2} + 4 (0)}{5 {(0)}^{2} - 25 \cdot 0 + 20}$ .

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Step 2.2.4.1

Simplify the numerator.

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Step 2.2.4.1.1

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

$y = \frac{- 4 \cdot 0 + 4 (0)}{5 \cdot 0^{2} - 25 \cdot 0 + 20}$

Step 2.2.4.1.2

Multiply $- 4$ by $0$ .

$y = \frac{0 + 4 (0)}{5 \cdot 0^{2} - 25 \cdot 0 + 20}$

Step 2.2.4.1.3

Multiply $4$ by $0$ .

$y = \frac{0 + 0}{5 \cdot 0^{2} - 25 \cdot 0 + 20}$

Step 2.2.4.1.4

Add $0$ and $0$ .

$y = \frac{0}{5 \cdot 0^{2} - 25 \cdot 0 + 20}$

Step 2.2.4.2

Simplify the denominator.

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Step 2.2.4.2.1

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

$y = \frac{0}{5 \cdot 0 - 25 \cdot 0 + 20}$

Step 2.2.4.2.2

Multiply $5$ by $0$ .

$y = \frac{0}{0 - 25 \cdot 0 + 20}$

Step 2.2.4.2.3

Multiply $- 25$ by $0$ .

$y = \frac{0}{0 + 0 + 20}$

Step 2.2.4.2.4

Add $0$ and $0$ .

$y = \frac{0}{0 + 20}$

Step 2.2.4.2.5

Add $0$ and $20$ .

$y = \frac{0}{20}$

Step 2.2.4.3

Divide $0$ by $20$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

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

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

Step 4