Algebra Examples

$\frac{z^{2} - 16 z + 64}{40 - 5 z} \cdot \frac{c}{z^{2} - 10 z + 16} = 1$

Step 1

Simplify $\frac{z^{2} - 16 z + 64}{40 - 5 z} \cdot \frac{c}{z^{2} - 10 z + 16}$ .

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

Rewrite $64$ as $8^{2}$ .

$\frac{z^{2} - 16 z + 8^{2}}{40 - 5 z} \cdot \frac{c}{z^{2} - 10 z + 16} = 1$

Step 1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$16 z = 2 \cdot z \cdot 8$

Step 1.1.3

Rewrite the polynomial.

$\frac{z^{2} - 2 \cdot z \cdot 8 + 8^{2}}{40 - 5 z} \cdot \frac{c}{z^{2} - 10 z + 16} = 1$

Step 1.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = z$ and $b = 8$ .

$\frac{{(z - 8)}^{2}}{40 - 5 z} \cdot \frac{c}{z^{2} - 10 z + 16} = 1$

Step 1.2

Factor $5$ out of $40 - 5 z$ .

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Step 1.2.1

Factor $5$ out of $40$ .

$\frac{{(z - 8)}^{2}}{5 (8) - 5 z} \cdot \frac{c}{z^{2} - 10 z + 16} = 1$

Step 1.2.2

Factor $5$ out of $- 5 z$ .

$\frac{{(z - 8)}^{2}}{5 (8) + 5 (- z)} \cdot \frac{c}{z^{2} - 10 z + 16} = 1$

Step 1.2.3

Factor $5$ out of $5 (8) + 5 (- z)$ .

$\frac{{(z - 8)}^{2}}{5 (8 - z)} \cdot \frac{c}{z^{2} - 10 z + 16} = 1$

Step 1.3

Factor $z^{2} - 10 z + 16$ using the AC method.

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Step 1.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $16$ and whose sum is $- 10$ .

$- 8, - 2$

Step 1.3.2

Write the factored form using these integers.

$\frac{{(z - 8)}^{2}}{5 (8 - z)} \cdot \frac{c}{(z - 8) (z - 2)} = 1$

Step 1.4

Simplify terms.

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Step 1.4.1

Combine.

$\frac{{(z - 8)}^{2} c}{5 (8 - z) ((z - 8) (z - 2))} = 1$

Step 1.4.2

Cancel the common factor of ${(z - 8)}^{2}$ and $z - 8$ .

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Step 1.4.2.1

Factor $z - 8$ out of ${(z - 8)}^{2} c$ .

$\frac{(z - 8) ((z - 8) c)}{5 (8 - z) ((z - 8) (z - 2))} = 1$

Step 1.4.2.2

Cancel the common factors.

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Step 1.4.2.2.1

Factor $z - 8$ out of $5 (8 - z) ((z - 8) (z - 2))$ .

$\frac{(z - 8) ((z - 8) c)}{(z - 8) (5 (8 - z) (z - 2))} = 1$

Step 1.4.2.2.2

Cancel the common factor.

$\frac{(z - 8) ((z - 8) c)}{(z - 8) (5 (8 - z) (z - 2))} = 1$

Step 1.4.2.2.3

Rewrite the expression.

$\frac{(z - 8) c}{5 (8 - z) (z - 2)} = 1$

Step 1.4.3

Cancel the common factor of $z - 8$ and $8 - z$ .

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Step 1.4.3.1

Factor $- 1$ out of $z$ .

$\frac{(- 1 (- z) - 8) c}{5 (8 - z) (z - 2)} = 1$

Step 1.4.3.2

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{(- 1 (- z) - 1 (8)) c}{5 (8 - z) (z - 2)} = 1$

Step 1.4.3.3

Factor $- 1$ out of $- 1 (- z) - 1 (8)$ .

$\frac{- 1 (- z + 8) c}{5 (8 - z) (z - 2)} = 1$

Step 1.4.3.4

Reorder terms.

$\frac{- 1 (- z + 8) c}{5 (- z + 8) (z - 2)} = 1$

Step 1.4.3.5

Cancel the common factor.

$\frac{- 1 (- z + 8) c}{5 (- z + 8) (z - 2)} = 1$

Step 1.4.3.6

Rewrite the expression.

$\frac{- 1 c}{5 (z - 2)} = 1$

Step 1.4.4

Move the negative in front of the fraction.

$- \frac{c}{5 (z - 2)} = 1$

Step 2

Divide each term in $- \frac{c}{5 (z - 2)} = 1$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- \frac{c}{5 (z - 2)} = 1$ by $- 1$ .

$\frac{- \frac{c}{5 (z - 2)}}{- 1} = \frac{1}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{c}{5 (z - 2)}}{1} = \frac{1}{- 1}$

Step 2.2.2

Divide $\frac{c}{5 (z - 2)}$ by $1$ .

$\frac{c}{5 (z - 2)} = \frac{1}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $1$ by $- 1$ .

$\frac{c}{5 (z - 2)} = - 1$

Step 3

Multiply both sides by $5 (z - 2)$ .

$\frac{c}{5 (z - 2)} (5 (z - 2)) = - (5 (z - 2))$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{c}{5 (z - 2)} (5 (z - 2))$ .

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Step 4.1.1.1

Rewrite using the commutative property of multiplication.

$5 \frac{c}{5 (z - 2)} (z - 2) = - (5 (z - 2))$

Step 4.1.1.2

Cancel the common factor of $5$ .

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Step 4.1.1.2.1

Cancel the common factor.

$5 \frac{c}{5 (z - 2)} (z - 2) = - (5 (z - 2))$

Step 4.1.1.2.2

Rewrite the expression.

$\frac{c}{z - 2} (z - 2) = - (5 (z - 2))$

Step 4.1.1.3

Cancel the common factor of $z - 2$ .

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Step 4.1.1.3.1

Cancel the common factor.

$\frac{c}{z - 2} (z - 2) = - (5 (z - 2))$

Step 4.1.1.3.2

Rewrite the expression.

$c = - (5 (z - 2))$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- (5 (z - 2))$ .

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Step 4.2.1.1

Apply the distributive property.

$c = - (5 z + 5 \cdot - 2)$

Step 4.2.1.2

Multiply $5$ by $- 2$ .

$c = - (5 z - 10)$

Step 4.2.1.3

Apply the distributive property.

$c = - (5 z) - - 10$

Step 4.2.1.4

Multiply.

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Step 4.2.1.4.1

Multiply $5$ by $- 1$ .

$c = - 5 z - - 10$

Step 4.2.1.4.2

Multiply $- 1$ by $- 10$ .

$c = - 5 z + 10$