Algebra Examples

$(8 z - 9) (8 z + 9) = a z^{2} - b$

Step 1

Rewrite the equation as $a z^{2} - b = (8 z - 9) (8 z + 9)$ .

$a z^{2} - b = (8 z - 9) (8 z + 9)$

Step 2

Simplify $(8 z - 9) (8 z + 9)$ .

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Step 2.1

Rewrite.

$a z^{2} - b = 0 + 0 + (8 z - 9) (8 z + 9)$

Step 2.2

Simplify by adding zeros.

$a z^{2} - b = (8 z - 9) (8 z + 9)$

Step 2.3

Expand $(8 z - 9) (8 z + 9)$ using the FOIL Method.

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Step 2.3.1

Apply the distributive property.

$a z^{2} - b = 8 z (8 z + 9) - 9 (8 z + 9)$

Step 2.3.2

Apply the distributive property.

$a z^{2} - b = 8 z (8 z) + 8 z \cdot 9 - 9 (8 z + 9)$

Step 2.3.3

Apply the distributive property.

$a z^{2} - b = 8 z (8 z) + 8 z \cdot 9 - 9 (8 z) - 9 \cdot 9$

Step 2.4

Simplify and combine like terms.

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

Rewrite using the commutative property of multiplication.

$a z^{2} - b = 8 \cdot 8 z \cdot z + 8 z \cdot 9 - 9 (8 z) - 9 \cdot 9$

Step 2.4.1.2

Multiply $z$ by $z$ by adding the exponents.

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Step 2.4.1.2.1

Move $z$ .

$a z^{2} - b = 8 \cdot 8 (z \cdot z) + 8 z \cdot 9 - 9 (8 z) - 9 \cdot 9$

Step 2.4.1.2.2

Multiply $z$ by $z$ .

$a z^{2} - b = 8 \cdot 8 z^{2} + 8 z \cdot 9 - 9 (8 z) - 9 \cdot 9$

Step 2.4.1.3

Multiply $8$ by $8$ .

$a z^{2} - b = 64 z^{2} + 8 z \cdot 9 - 9 (8 z) - 9 \cdot 9$

Step 2.4.1.4

Multiply $9$ by $8$ .

$a z^{2} - b = 64 z^{2} + 72 z - 9 (8 z) - 9 \cdot 9$

Step 2.4.1.5

Multiply $8$ by $- 9$ .

$a z^{2} - b = 64 z^{2} + 72 z - 72 z - 9 \cdot 9$

Step 2.4.1.6

Multiply $- 9$ by $9$ .

$a z^{2} - b = 64 z^{2} + 72 z - 72 z - 81$

Step 2.4.2

Subtract $72 z$ from $72 z$ .

$a z^{2} - b = 64 z^{2} + 0 - 81$

Step 2.4.3

Add $64 z^{2}$ and $0$ .

$a z^{2} - b = 64 z^{2} - 81$

Step 3

Add $b$ to both sides of the equation.

$a z^{2} = 64 z^{2} - 81 + b$

Step 4

Divide each term in $a z^{2} = 64 z^{2} - 81 + b$ by $z^{2}$ and simplify.

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Step 4.1

Divide each term in $a z^{2} = 64 z^{2} - 81 + b$ by $z^{2}$ .

$\frac{a z^{2}}{z^{2}} = \frac{64 z^{2}}{z^{2}} + \frac{- 81}{z^{2}} + \frac{b}{z^{2}}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $z^{2}$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{a z^{2}}{z^{2}} = \frac{64 z^{2}}{z^{2}} + \frac{- 81}{z^{2}} + \frac{b}{z^{2}}$

Step 4.2.1.2

Divide $a$ by $1$ .

$a = \frac{64 z^{2}}{z^{2}} + \frac{- 81}{z^{2}} + \frac{b}{z^{2}}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Cancel the common factor of $z^{2}$ .

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Step 4.3.1.1.1

Cancel the common factor.

$a = \frac{64 z^{2}}{z^{2}} + \frac{- 81}{z^{2}} + \frac{b}{z^{2}}$

Step 4.3.1.1.2

Divide $64$ by $1$ .

$a = 64 + \frac{- 81}{z^{2}} + \frac{b}{z^{2}}$

Step 4.3.1.2

Move the negative in front of the fraction.

$a = 64 - \frac{81}{z^{2}} + \frac{b}{z^{2}}$