Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

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

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

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

$2 x - 2 = 0$

$x + 2 = 0$

Step 1.2.2.2

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

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Step 1.2.2.2.1

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

$2 x - 2 = 0$

Step 1.2.2.2.2

Solve $2 x - 2 = 0$ for $x$ .

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Step 1.2.2.2.2.1

Add $2$ to both sides of the equation.

$2 x = 2$

Step 1.2.2.2.2.2

Divide each term in $2 x = 2$ by $2$ and simplify.

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Step 1.2.2.2.2.2.1

Divide each term in $2 x = 2$ by $2$ .

$\frac{2 x}{2} = \frac{2}{2}$

Step 1.2.2.2.2.2.2

Simplify the left side.

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Step 1.2.2.2.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.2.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{2}{2}$

Step 1.2.2.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{2}$

Step 1.2.2.2.2.2.3

Simplify the right side.

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Step 1.2.2.2.2.2.3.1

Divide $2$ by $2$ .

$x = 1$

Step 1.2.2.3

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

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Step 1.2.2.3.1

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

$x + 2 = 0$

Step 1.2.2.3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.2.4

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

$x = 1, - 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(1, 0), (- 2, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Simplify $\frac{(2 (0) - 2) ((0) + 2)}{(4 (0) + 4) (2 (0) + 1)}$ .

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Step 2.2.3.1

Reduce the expression by cancelling the common factors.

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Step 2.2.3.1.1

Cancel the common factor of $(0) + 2$ and $4 (0) + 4$ .

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Step 2.2.3.1.1.1

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

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

Step 2.2.3.1.1.2

Cancel the common factors.

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Step 2.2.3.1.1.2.1

Factor $2$ out of $(4 (0) + 4) (2 (0) + 1)$ .

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

Step 2.2.3.1.1.2.2

Cancel the common factor.

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

Step 2.2.3.1.1.2.3

Rewrite the expression.

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

Step 2.2.3.1.2

Cancel the common factor of $(0) + 2$ and $2 \cdot 0 + 2$ .

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Step 2.2.3.1.2.1

Factor $2$ out of $(0 - 1) ((0) + 2)$ .

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

Step 2.2.3.1.2.2

Cancel the common factors.

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Step 2.2.3.1.2.2.1

Factor $2$ out of $(2 \cdot 0 + 2) (2 (0) + 1)$ .

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

Step 2.2.3.1.2.2.2

Cancel the common factor.

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

Step 2.2.3.1.2.2.3

Rewrite the expression.

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

Step 2.2.3.1.3

Cancel the common factor of $0 + 1$ .

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Step 2.2.3.1.3.1

Cancel the common factor.

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

Step 2.2.3.1.3.2

Rewrite the expression.

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

Step 2.2.3.1.4

Subtract $1$ from $0$ .

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

Step 2.2.3.2

Simplify the denominator.

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Step 2.2.3.2.1

Multiply $2$ by $0$ .

$y = \frac{- 1}{0 + 1}$

Step 2.2.3.2.2

Add $0$ and $1$ .

$y = \frac{- 1}{1}$

Step 2.2.3.3

Divide $- 1$ by $1$ .

$y = - 1$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 1)$

Step 3

List the intersections.

x-intercept(s): $(1, 0), (- 2, 0)$

y-intercept(s): $(0, - 1)$

Step 4