Algebra Examples

$f (x) = \frac{x^{2} - 4 x + 4}{x^{3} - 5 x^{2}}$

Step 1

Substitute $- 2$ for $x$ and find the result for $y$ .

$y = \frac{{(- 2)}^{2} - 4 \cdot - 2 + 4}{{(- 2)}^{3} - 5 {(- 2)}^{2}}$

Step 2

Solve the equation for $y$ .

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Step 2.1

Remove parentheses.

$y = \frac{{(- 2)}^{2} - 4 \cdot - 2 + 4}{{(- 2)}^{3} - 5 {(- 2)}^{2}}$

Step 2.2

Simplify $\frac{{(- 2)}^{2} - 4 \cdot - 2 + 4}{{(- 2)}^{3} - 5 {(- 2)}^{2}}$ .

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

Raise $- 2$ to the power of $2$ .

$y = \frac{4 - 4 \cdot - 2 + 4}{{(- 2)}^{3} - 5 {(- 2)}^{2}}$

Step 2.2.1.2

Multiply $- 4$ by $- 2$ .

$y = \frac{4 + 8 + 4}{{(- 2)}^{3} - 5 {(- 2)}^{2}}$

Step 2.2.1.3

Add $4$ and $8$ .

$y = \frac{12 + 4}{{(- 2)}^{3} - 5 {(- 2)}^{2}}$

Step 2.2.1.4

Add $12$ and $4$ .

$y = \frac{16}{{(- 2)}^{3} - 5 {(- 2)}^{2}}$

Step 2.2.2

Simplify the denominator.

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Step 2.2.2.1

Raise $- 2$ to the power of $3$ .

$y = \frac{16}{- 8 - 5 {(- 2)}^{2}}$

Step 2.2.2.2

Raise $- 2$ to the power of $2$ .

$y = \frac{16}{- 8 - 5 \cdot 4}$

Step 2.2.2.3

Multiply $- 5$ by $4$ .

$y = \frac{16}{- 8 - 20}$

Step 2.2.2.4

Subtract $20$ from $- 8$ .

$y = \frac{16}{- 28}$

Step 2.2.3

Reduce the expression by cancelling the common factors.

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Step 2.2.3.1

Cancel the common factor of $16$ and $- 28$ .

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Step 2.2.3.1.1

Factor $4$ out of $16$ .

$y = \frac{4 (4)}{- 28}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $4$ out of $- 28$ .

$y = \frac{4 \cdot 4}{4 \cdot - 7}$

Step 2.2.3.1.2.2

Cancel the common factor.

$y = \frac{4 \cdot 4}{4 \cdot - 7}$

Step 2.2.3.1.2.3

Rewrite the expression.

$y = \frac{4}{- 7}$

Step 2.2.3.2

Move the negative in front of the fraction.

$y = - \frac{4}{7}$

Step 3

Substitute $- 1$ for $x$ and find the result for $y$ .

$y = \frac{{(- 1)}^{2} - 4 \cdot - 1 + 4}{{(- 1)}^{3} - 5 {(- 1)}^{2}}$

Step 4

Solve the equation for $y$ .

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Step 4.1

Remove parentheses.

$y = \frac{{(- 1)}^{2} - 4 \cdot - 1 + 4}{{(- 1)}^{3} - 5 {(- 1)}^{2}}$

Step 4.2

Simplify $\frac{{(- 1)}^{2} - 4 \cdot - 1 + 4}{{(- 1)}^{3} - 5 {(- 1)}^{2}}$ .

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Step 4.2.1

Simplify the numerator.

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Step 4.2.1.1

Raise $- 1$ to the power of $2$ .

$y = \frac{1 - 4 \cdot - 1 + 4}{{(- 1)}^{3} - 5 {(- 1)}^{2}}$

Step 4.2.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{1 + 4 + 4}{{(- 1)}^{3} - 5 {(- 1)}^{2}}$

Step 4.2.1.3

Add $1$ and $4$ .

$y = \frac{5 + 4}{{(- 1)}^{3} - 5 {(- 1)}^{2}}$

Step 4.2.1.4

Add $5$ and $4$ .

$y = \frac{9}{{(- 1)}^{3} - 5 {(- 1)}^{2}}$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

Raise $- 1$ to the power of $3$ .

$y = \frac{9}{- 1 - 5 {(- 1)}^{2}}$

Step 4.2.2.2

Raise $- 1$ to the power of $2$ .

$y = \frac{9}{- 1 - 5 \cdot 1}$

Step 4.2.2.3

Multiply $- 5$ by $1$ .

$y = \frac{9}{- 1 - 5}$

Step 4.2.2.4

Subtract $5$ from $- 1$ .

$y = \frac{9}{- 6}$

Step 4.2.3

Reduce the expression by cancelling the common factors.

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Step 4.2.3.1

Cancel the common factor of $9$ and $- 6$ .

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Step 4.2.3.1.1

Factor $3$ out of $9$ .

$y = \frac{3 (3)}{- 6}$

Step 4.2.3.1.2

Cancel the common factors.

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Step 4.2.3.1.2.1

Factor $3$ out of $- 6$ .

$y = \frac{3 \cdot 3}{3 \cdot - 2}$

Step 4.2.3.1.2.2

Cancel the common factor.

$y = \frac{3 \cdot 3}{3 \cdot - 2}$

Step 4.2.3.1.2.3

Rewrite the expression.

$y = \frac{3}{- 2}$

Step 4.2.3.2

Move the negative in front of the fraction.

$y = - \frac{3}{2}$

Step 5

Substitute $1$ for $x$ and find the result for $y$ .

$y = \frac{{(1)}^{2} - 4 \cdot 1 + 4}{{(1)}^{3} - 5 {(1)}^{2}}$

Step 6

Solve the equation for $y$ .

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Step 6.1

Remove parentheses.

$y = \frac{1^{2} - 4 \cdot 1 + 4}{{(1)}^{3} - 5 {(1)}^{2}}$

Step 6.2

Remove parentheses.

$y = \frac{1^{2} - 4 \cdot 1 + 4}{1^{3} - 5 {(1)}^{2}}$

Step 6.3

Remove parentheses.

$y = \frac{1^{2} - 4 \cdot 1 + 4}{1^{3} - 5 \cdot 1^{2}}$

Step 6.4

Remove parentheses.

$y = \frac{{(1)}^{2} - 4 \cdot 1 + 4}{{(1)}^{3} - 5 {(1)}^{2}}$

Step 6.5

Simplify $\frac{{(1)}^{2} - 4 \cdot 1 + 4}{{(1)}^{3} - 5 {(1)}^{2}}$ .

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Step 6.5.1

Simplify the numerator.

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Step 6.5.1.1

One to any power is one.

$y = \frac{1 - 4 \cdot 1 + 4}{1^{3} - 5 \cdot 1^{2}}$

Step 6.5.1.2

Multiply $- 4$ by $1$ .

$y = \frac{1 - 4 + 4}{1^{3} - 5 \cdot 1^{2}}$

Step 6.5.1.3

Subtract $4$ from $1$ .

$y = \frac{- 3 + 4}{1^{3} - 5 \cdot 1^{2}}$

Step 6.5.1.4

Add $- 3$ and $4$ .

$y = \frac{1}{1^{3} - 5 \cdot 1^{2}}$

Step 6.5.2

Simplify the denominator.

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Step 6.5.2.1

One to any power is one.

$y = \frac{1}{1 - 5 \cdot 1^{2}}$

Step 6.5.2.2

One to any power is one.

$y = \frac{1}{1 - 5 \cdot 1}$

Step 6.5.2.3

Multiply $- 5$ by $1$ .

$y = \frac{1}{1 - 5}$

Step 6.5.2.4

Subtract $5$ from $1$ .

$y = \frac{1}{- 4}$

Step 6.5.3

Move the negative in front of the fraction.

$y = - \frac{1}{4}$

Step 7

Substitute $2$ for $x$ and find the result for $y$ .

$y = \frac{{(2)}^{2} - 4 \cdot 2 + 4}{{(2)}^{3} - 5 {(2)}^{2}}$

Step 8

Solve the equation for $y$ .

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Step 8.1

Remove parentheses.

$y = \frac{2^{2} - 4 \cdot 2 + 4}{{(2)}^{3} - 5 {(2)}^{2}}$

Step 8.2

Remove parentheses.

$y = \frac{2^{2} - 4 \cdot 2 + 4}{2^{3} - 5 {(2)}^{2}}$

Step 8.3

Remove parentheses.

$y = \frac{2^{2} - 4 \cdot 2 + 4}{2^{3} - 5 \cdot 2^{2}}$

Step 8.4

Remove parentheses.

$y = \frac{{(2)}^{2} - 4 \cdot 2 + 4}{{(2)}^{3} - 5 {(2)}^{2}}$

Step 8.5

Simplify $\frac{{(2)}^{2} - 4 \cdot 2 + 4}{{(2)}^{3} - 5 {(2)}^{2}}$ .

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Step 8.5.1

Simplify the numerator.

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Step 8.5.1.1

Raise $2$ to the power of $2$ .

$y = \frac{4 - 4 \cdot 2 + 4}{2^{3} - 5 \cdot 2^{2}}$

Step 8.5.1.2

Multiply $- 4$ by $2$ .

$y = \frac{4 - 8 + 4}{2^{3} - 5 \cdot 2^{2}}$

Step 8.5.1.3

Subtract $8$ from $4$ .

$y = \frac{- 4 + 4}{2^{3} - 5 \cdot 2^{2}}$

Step 8.5.1.4

Add $- 4$ and $4$ .

$y = \frac{0}{2^{3} - 5 \cdot 2^{2}}$

Step 8.5.2

Simplify the denominator.

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Step 8.5.2.1

Raise $2$ to the power of $3$ .

$y = \frac{0}{8 - 5 \cdot 2^{2}}$

Step 8.5.2.2

Raise $2$ to the power of $2$ .

$y = \frac{0}{8 - 5 \cdot 4}$

Step 8.5.2.3

Multiply $- 5$ by $4$ .

$y = \frac{0}{8 - 20}$

Step 8.5.2.4

Subtract $20$ from $8$ .

$y = \frac{0}{- 12}$

Step 8.5.3

Divide $0$ by $- 12$ .

$y = 0$

Step 9

This is a table of possible values to use when graphing the equation.

$\begin{array}{c} x & y \\ - 2 & - \frac{4}{7} \\ - 1 & - \frac{3}{2} \\ 1 & - \frac{1}{4} \\ 2 & 0 \end{array}$

Step 10