Finite Math Examples

$\frac{z}{z - 3} + \frac{3}{z + 3} = \frac{18}{z^{2 - 9}}$

Step 1

Subtract $\frac{18}{z^{2 - 9}}$ from both sides of the equation.

$\frac{z}{z - 3} + \frac{3}{z + 3} - \frac{18}{z^{2 - 9}} = 0$

Step 2

Simplify $\frac{z}{z - 3} + \frac{3}{z + 3} - \frac{18}{z^{2 - 9}}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Move $z^{2 - 9}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{z}{z - 3} + \frac{3}{z + 3} - (18 z^{- (2 - 9)}) = 0$

Step 2.1.2

Subtract $9$ from $2$ .

$\frac{z}{z - 3} + \frac{3}{z + 3} - (18 z^{- - 7}) = 0$

Step 2.1.3

Multiply $- 1$ by $- 7$ .

$\frac{z}{z - 3} + \frac{3}{z + 3} - (18 z^{7}) = 0$

Step 2.1.4

Multiply $18$ by $- 1$ .

$\frac{z}{z - 3} + \frac{3}{z + 3} - 18 z^{7} = 0$

Step 2.2

To write $\frac{z}{z - 3}$ as a fraction with a common denominator, multiply by $\frac{z + 3}{z + 3}$ .

$\frac{z}{z - 3} \cdot \frac{z + 3}{z + 3} + \frac{3}{z + 3} - 18 z^{7} = 0$

Step 2.3

To write $\frac{3}{z + 3}$ as a fraction with a common denominator, multiply by $\frac{z - 3}{z - 3}$ .

$\frac{z}{z - 3} \cdot \frac{z + 3}{z + 3} + \frac{3}{z + 3} \cdot \frac{z - 3}{z - 3} - 18 z^{7} = 0$

Step 2.4

Write each expression with a common denominator of $(z - 3) (z + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{z}{z - 3}$ by $\frac{z + 3}{z + 3}$ .

$\frac{z (z + 3)}{(z - 3) (z + 3)} + \frac{3}{z + 3} \cdot \frac{z - 3}{z - 3} - 18 z^{7} = 0$

Step 2.4.2

Multiply $\frac{3}{z + 3}$ by $\frac{z - 3}{z - 3}$ .

$\frac{z (z + 3)}{(z - 3) (z + 3)} + \frac{3 (z - 3)}{(z + 3) (z - 3)} - 18 z^{7} = 0$

Step 2.4.3

Reorder the factors of $(z - 3) (z + 3)$ .

$\frac{z (z + 3)}{(z + 3) (z - 3)} + \frac{3 (z - 3)}{(z + 3) (z - 3)} - 18 z^{7} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{z (z + 3) + 3 (z - 3)}{(z + 3) (z - 3)} - 18 z^{7} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Apply the distributive property.

$\frac{z \cdot z + z \cdot 3 + 3 (z - 3)}{(z + 3) (z - 3)} - 18 z^{7} = 0$

Step 2.6.2

Multiply $z$ by $z$ .

$\frac{z^{2} + z \cdot 3 + 3 (z - 3)}{(z + 3) (z - 3)} - 18 z^{7} = 0$

Step 2.6.3

Move $3$ to the left of $z$ .

$\frac{z^{2} + 3 \cdot z + 3 (z - 3)}{(z + 3) (z - 3)} - 18 z^{7} = 0$

Step 2.6.4

Apply the distributive property.

$\frac{z^{2} + 3 z + 3 z + 3 \cdot - 3}{(z + 3) (z - 3)} - 18 z^{7} = 0$

Step 2.6.5

Multiply $3$ by $- 3$ .

$\frac{z^{2} + 3 z + 3 z - 9}{(z + 3) (z - 3)} - 18 z^{7} = 0$

Step 2.6.6

Add $3 z$ and $3 z$ .

$\frac{z^{2} + 6 z - 9}{(z + 3) (z - 3)} - 18 z^{7} = 0$

Step 2.7

To write $- 18 z^{7}$ as a fraction with a common denominator, multiply by $\frac{(z + 3) (z - 3)}{(z + 3) (z - 3)}$ .

$\frac{z^{2} + 6 z - 9}{(z + 3) (z - 3)} - 18 z^{7} \cdot \frac{(z + 3) (z - 3)}{(z + 3) (z - 3)} = 0$

Step 2.8

Combine $- 18 z^{7}$ and $\frac{(z + 3) (z - 3)}{(z + 3) (z - 3)}$ .

$\frac{z^{2} + 6 z - 9}{(z + 3) (z - 3)} + \frac{- 18 z^{7} ((z + 3) (z - 3))}{(z + 3) (z - 3)} = 0$

Step 2.9

Combine the numerators over the common denominator.

$\frac{z^{2} + 6 z - 9 - 18 z^{7} ((z + 3) (z - 3))}{(z + 3) (z - 3)} = 0$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Expand $(z + 3) (z - 3)$ using the FOIL Method.

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Step 2.10.1.1

Apply the distributive property.

$\frac{z^{2} + 6 z - 9 - 18 z^{7} (z (z - 3) + 3 (z - 3))}{(z + 3) (z - 3)} = 0$

Step 2.10.1.2

Apply the distributive property.

$\frac{z^{2} + 6 z - 9 - 18 z^{7} (z \cdot z + z \cdot - 3 + 3 (z - 3))}{(z + 3) (z - 3)} = 0$

Step 2.10.1.3

Apply the distributive property.

$\frac{z^{2} + 6 z - 9 - 18 z^{7} (z \cdot z + z \cdot - 3 + 3 z + 3 \cdot - 3)}{(z + 3) (z - 3)} = 0$

Step 2.10.2

Combine the opposite terms in $z \cdot z + z \cdot - 3 + 3 z + 3 \cdot - 3$ .

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Step 2.10.2.1

Reorder the factors in the terms $z \cdot - 3$ and $3 z$ .

$\frac{z^{2} + 6 z - 9 - 18 z^{7} (z \cdot z - 3 z + 3 z + 3 \cdot - 3)}{(z + 3) (z - 3)} = 0$

Step 2.10.2.2

Add $- 3 z$ and $3 z$ .

$\frac{z^{2} + 6 z - 9 - 18 z^{7} (z \cdot z + 0 + 3 \cdot - 3)}{(z + 3) (z - 3)} = 0$

Step 2.10.2.3

Add $z \cdot z$ and $0$ .

$\frac{z^{2} + 6 z - 9 - 18 z^{7} (z \cdot z + 3 \cdot - 3)}{(z + 3) (z - 3)} = 0$

Step 2.10.3

Simplify each term.

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Step 2.10.3.1

Multiply $z$ by $z$ .

$\frac{z^{2} + 6 z - 9 - 18 z^{7} (z^{2} + 3 \cdot - 3)}{(z + 3) (z - 3)} = 0$

Step 2.10.3.2

Multiply $3$ by $- 3$ .

$\frac{z^{2} + 6 z - 9 - 18 z^{7} (z^{2} - 9)}{(z + 3) (z - 3)} = 0$

Step 2.10.4

Apply the distributive property.

$\frac{z^{2} + 6 z - 9 - 18 z^{7} z^{2} - 18 z^{7} \cdot - 9}{(z + 3) (z - 3)} = 0$

Step 2.10.5

Multiply $z^{7}$ by $z^{2}$ by adding the exponents.

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Step 2.10.5.1

Move $z^{2}$ .

$\frac{z^{2} + 6 z - 9 - 18 (z^{2} z^{7}) - 18 z^{7} \cdot - 9}{(z + 3) (z - 3)} = 0$

Step 2.10.5.2

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

$\frac{z^{2} + 6 z - 9 - 18 z^{2 + 7} - 18 z^{7} \cdot - 9}{(z + 3) (z - 3)} = 0$

Step 2.10.5.3

Add $2$ and $7$ .

$\frac{z^{2} + 6 z - 9 - 18 z^{9} - 18 z^{7} \cdot - 9}{(z + 3) (z - 3)} = 0$

Step 2.10.6

Multiply $- 9$ by $- 18$ .

$\frac{z^{2} + 6 z - 9 - 18 z^{9} + 162 z^{7}}{(z + 3) (z - 3)} = 0$

Step 2.10.7

Reorder terms.

$\frac{- 18 z^{9} + 162 z^{7} + z^{2} + 6 z - 9}{(z + 3) (z - 3)} = 0$

Step 2.11

Factor $- 1$ out of $- 18 z^{9}$ .

$\frac{- (18 z^{9}) + 162 z^{7} + z^{2} + 6 z - 9}{(z + 3) (z - 3)} = 0$

Step 2.12

Factor $- 1$ out of $162 z^{7}$ .

$\frac{- (18 z^{9}) - (- 162 z^{7}) + z^{2} + 6 z - 9}{(z + 3) (z - 3)} = 0$

Step 2.13

Factor $- 1$ out of $- (18 z^{9}) - (- 162 z^{7})$ .

$\frac{- (18 z^{9} - 162 z^{7}) + z^{2} + 6 z - 9}{(z + 3) (z - 3)} = 0$

Step 2.14

Factor $- 1$ out of $z^{2}$ .

$\frac{- (18 z^{9} - 162 z^{7}) - 1 (- z^{2}) + 6 z - 9}{(z + 3) (z - 3)} = 0$

Step 2.15

Factor $- 1$ out of $- (18 z^{9} - 162 z^{7}) - 1 (- z^{2})$ .

$\frac{- (18 z^{9} - 162 z^{7} - z^{2}) + 6 z - 9}{(z + 3) (z - 3)} = 0$

Step 2.16

Factor $- 1$ out of $6 z$ .

$\frac{- (18 z^{9} - 162 z^{7} - z^{2}) - (- 6 z) - 9}{(z + 3) (z - 3)} = 0$

Step 2.17

Factor $- 1$ out of $- (18 z^{9} - 162 z^{7} - z^{2}) - (- 6 z)$ .

$\frac{- (18 z^{9} - 162 z^{7} - z^{2} - 6 z) - 9}{(z + 3) (z - 3)} = 0$

Step 2.18

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{- (18 z^{9} - 162 z^{7} - z^{2} - 6 z) - 1 (9)}{(z + 3) (z - 3)} = 0$

Step 2.19

Factor $- 1$ out of $- (18 z^{9} - 162 z^{7} - z^{2} - 6 z) - 1 (9)$ .

$\frac{- (18 z^{9} - 162 z^{7} - z^{2} - 6 z + 9)}{(z + 3) (z - 3)} = 0$

Step 2.20

Rewrite $- (18 z^{9} - 162 z^{7} - z^{2} - 6 z + 9)$ as $- 1 (18 z^{9} - 162 z^{7} - z^{2} - 6 z + 9)$ .

$\frac{- 1 (18 z^{9} - 162 z^{7} - z^{2} - 6 z + 9)}{(z + 3) (z - 3)} = 0$

Step 2.21

Move the negative in front of the fraction.

$- \frac{18 z^{9} - 162 z^{7} - z^{2} - 6 z + 9}{(z + 3) (z - 3)} = 0$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$z \approx 0.6113853$

Step 4