Algebra Examples

$f (x) = \frac{- 3 x (x - 2)}{(x - 2) (x + 2)}$

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 $x$ .

$0 = \frac{- 3 x (x - 2)}{(x - 2) (x + 2)}$

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

$- 3 x (x - 2) = 0$

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 $0$ .

$x = 0$

$x - 2 = 0$

Step 1.2.2.2

Set $x$ equal to $0$ .

$x = 0$

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

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.2.4

The final solution is all the values that make $- 3 x (x - 2) = 0$ true.

$x = 0, 2$

Step 1.2.3

Exclude the solutions that do not make $0 = \frac{- 3 x (x - 2)}{(x - 2) (x + 2)}$ 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 $y$ .

$y = \frac{- 3 \cdot 0 ((0) - 2)}{((0) - 2) ((0) + 2)}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{- 3 \cdot (0 (0 - 2))}{((0) - 2) ((0) + 2)}$

Step 2.2.2

Remove parentheses.

$y = \frac{- 3 \cdot (0 (0 - 2))}{(0 - 2) ((0) + 2)}$

Step 2.2.3

Remove parentheses.

$y = \frac{- 3 \cdot (0 (0 - 2))}{(0 - 2) (0 + 2)}$

Step 2.2.4

Remove parentheses.

$y = \frac{- 3 \cdot (0 ((0) - 2))}{((0) - 2) ((0) + 2)}$

Step 2.2.5

Simplify $\frac{- 3 \cdot (0 ((0) - 2))}{((0) - 2) ((0) + 2)}$ .

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Step 2.2.5.1

Reduce the expression by cancelling the common factors.

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Step 2.2.5.1.1

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

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Step 2.2.5.1.1.1

Factor $2$ out of $- 3 \cdot (0 ((0) - 2))$ .

$y = \frac{2 (- 3 \cdot (0 ((0) - 2)))}{((0) - 2) ((0) + 2)}$

Step 2.2.5.1.1.2

Cancel the common factors.

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Step 2.2.5.1.1.2.1

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

$y = \frac{2 (- 3 \cdot (0 ((0) - 2)))}{2 ((0 - 1) ((0) + 2))}$

Step 2.2.5.1.1.2.2

Cancel the common factor.

$y = \frac{2 (- 3 \cdot (0 ((0) - 2)))}{2 ((0 - 1) ((0) + 2))}$

Step 2.2.5.1.1.2.3

Rewrite the expression.

$y = \frac{- 3 \cdot (0 ((0) - 2))}{(0 - 1) ((0) + 2)}$

Step 2.2.5.1.2

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

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Step 2.2.5.1.2.1

Factor $2$ out of $- 3 \cdot (0 ((0) - 2))$ .

$y = \frac{2 (- 3 \cdot (0 ((0) - 2)))}{(0 - 1) ((0) + 2)}$

Step 2.2.5.1.2.2

Cancel the common factors.

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Step 2.2.5.1.2.2.1

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

$y = \frac{2 (- 3 \cdot (0 ((0) - 2)))}{2 ((0 - 1) (0 + 1))}$

Step 2.2.5.1.2.2.2

Cancel the common factor.

$y = \frac{2 (- 3 \cdot (0 ((0) - 2)))}{2 ((0 - 1) (0 + 1))}$

Step 2.2.5.1.2.2.3

Rewrite the expression.

$y = \frac{- 3 \cdot (0 ((0) - 2))}{(0 - 1) (0 + 1)}$

Step 2.2.5.2

Simplify the numerator.

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Step 2.2.5.2.1

Multiply $- 3$ by $0$ .

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

Step 2.2.5.2.2

Multiply $0$ by $0 - 2$ .

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

Step 2.2.5.3

Simplify the denominator.

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Step 2.2.5.3.1

Rewrite $0$ as $- 1 (0)$ .

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

Step 2.2.5.3.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 2.2.5.3.3

Factor $- 1$ out of $- 1 \cdot 0 - 1 \cdot 1$ .

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

Step 2.2.5.3.4

Raise $0 + 1$ to the power of $1$ .

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

Step 2.2.5.3.5

Raise $0 + 1$ to the power of $1$ .

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

Step 2.2.5.3.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.5.3.7

Add $1$ and $1$ .

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

Step 2.2.5.4

Simplify the denominator.

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Step 2.2.5.4.1

Add $0$ and $1$ .

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

Step 2.2.5.4.2

One to any power is one.

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

Step 2.2.5.5

Simplify the expression.

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Step 2.2.5.5.1

Multiply $- 1$ by $1$ .

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

Step 2.2.5.5.2

Divide $0$ by $- 1$ .

$y = 0$

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