Finite Math Examples

${(3 m)}^{2} + {(2 m + 16)}^{2} = {(5 m)}^{2}$

Step 1

Subtract ${(5 m)}^{2}$ from both sides of the equation.

${(3 m)}^{2} + {(2 m + 16)}^{2} - {(5 m)}^{2} = 0$

Step 2

Simplify ${(3 m)}^{2} + {(2 m + 16)}^{2} - {(5 m)}^{2}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the product rule to $3 m$ .

$3^{2} m^{2} + {(2 m + 16)}^{2} - {(5 m)}^{2} = 0$

Step 2.1.2

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

$9 m^{2} + {(2 m + 16)}^{2} - {(5 m)}^{2} = 0$

Step 2.1.3

Rewrite ${(2 m + 16)}^{2}$ as $(2 m + 16) (2 m + 16)$ .

$9 m^{2} + (2 m + 16) (2 m + 16) - {(5 m)}^{2} = 0$

Step 2.1.4

Expand $(2 m + 16) (2 m + 16)$ using the FOIL Method.

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Step 2.1.4.1

Apply the distributive property.

$9 m^{2} + 2 m (2 m + 16) + 16 (2 m + 16) - {(5 m)}^{2} = 0$

Step 2.1.4.2

Apply the distributive property.

$9 m^{2} + 2 m (2 m) + 2 m \cdot 16 + 16 (2 m + 16) - {(5 m)}^{2} = 0$

Step 2.1.4.3

Apply the distributive property.

$9 m^{2} + 2 m (2 m) + 2 m \cdot 16 + 16 (2 m) + 16 \cdot 16 - {(5 m)}^{2} = 0$

Step 2.1.5

Simplify and combine like terms.

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Step 2.1.5.1

Simplify each term.

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Step 2.1.5.1.1

Rewrite using the commutative property of multiplication.

$9 m^{2} + 2 \cdot 2 m \cdot m + 2 m \cdot 16 + 16 (2 m) + 16 \cdot 16 - {(5 m)}^{2} = 0$

Step 2.1.5.1.2

Multiply $m$ by $m$ by adding the exponents.

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Step 2.1.5.1.2.1

Move $m$ .

$9 m^{2} + 2 \cdot 2 (m \cdot m) + 2 m \cdot 16 + 16 (2 m) + 16 \cdot 16 - {(5 m)}^{2} = 0$

Step 2.1.5.1.2.2

Multiply $m$ by $m$ .

$9 m^{2} + 2 \cdot 2 m^{2} + 2 m \cdot 16 + 16 (2 m) + 16 \cdot 16 - {(5 m)}^{2} = 0$

Step 2.1.5.1.3

Multiply $2$ by $2$ .

$9 m^{2} + 4 m^{2} + 2 m \cdot 16 + 16 (2 m) + 16 \cdot 16 - {(5 m)}^{2} = 0$

Step 2.1.5.1.4

Multiply $16$ by $2$ .

$9 m^{2} + 4 m^{2} + 32 m + 16 (2 m) + 16 \cdot 16 - {(5 m)}^{2} = 0$

Step 2.1.5.1.5

Multiply $2$ by $16$ .

$9 m^{2} + 4 m^{2} + 32 m + 32 m + 16 \cdot 16 - {(5 m)}^{2} = 0$

Step 2.1.5.1.6

Multiply $16$ by $16$ .

$9 m^{2} + 4 m^{2} + 32 m + 32 m + 256 - {(5 m)}^{2} = 0$

Step 2.1.5.2

Add $32 m$ and $32 m$ .

$9 m^{2} + 4 m^{2} + 64 m + 256 - {(5 m)}^{2} = 0$

Step 2.1.6

Apply the product rule to $5 m$ .

$9 m^{2} + 4 m^{2} + 64 m + 256 - (5^{2} m^{2}) = 0$

Step 2.1.7

Raise $5$ to the power of $2$ .

$9 m^{2} + 4 m^{2} + 64 m + 256 - (25 m^{2}) = 0$

Step 2.1.8

Multiply $25$ by $- 1$ .

$9 m^{2} + 4 m^{2} + 64 m + 256 - 25 m^{2} = 0$

Step 2.2

Add $9 m^{2}$ and $4 m^{2}$ .

$13 m^{2} + 64 m + 256 - 25 m^{2} = 0$

Step 2.3

Subtract $25 m^{2}$ from $13 m^{2}$ .

$- 12 m^{2} + 64 m + 256 = 0$

Step 3

Factor $- 4$ out of $- 12 m^{2} + 64 m + 256$ .

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Step 3.1

Factor $- 4$ out of $- 12 m^{2}$ .

$- 4 (3 m^{2}) + 64 m + 256 = 0$

Step 3.2

Factor $- 4$ out of $64 m$ .

$- 4 (3 m^{2}) - 4 (- 16 m) + 256 = 0$

Step 3.3

Factor $- 4$ out of $256$ .

$- 4 (3 m^{2}) - 4 (- 16 m) - 4 \cdot - 64 = 0$

Step 3.4

Factor $- 4$ out of $- 4 (3 m^{2}) - 4 (- 16 m)$ .

$- 4 (3 m^{2} - 16 m) - 4 \cdot - 64 = 0$

Step 3.5

Factor $- 4$ out of $- 4 (3 m^{2} - 16 m) - 4 (- 64)$ .

$- 4 (3 m^{2} - 16 m - 64) = 0$

Step 4

Factor.

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Step 4.1

Factor by grouping.

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Step 4.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 64 = - 192$ and whose sum is $b = - 16$ .

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Step 4.1.1.1

Factor $- 16$ out of $- 16 m$ .

$- 4 (3 m^{2} - 16 m - 64) = 0$

Step 4.1.1.2

Rewrite $- 16$ as $8$ plus $- 24$

$- 4 (3 m^{2} + (8 - 24) m - 64) = 0$

Step 4.1.1.3

Apply the distributive property.

$- 4 (3 m^{2} + 8 m - 24 m - 64) = 0$

Step 4.1.2

Factor out the greatest common factor from each group.

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Step 4.1.2.1

Group the first two terms and the last two terms.

$- 4 ((3 m^{2} + 8 m) - 24 m - 64) = 0$

Step 4.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- 4 (m (3 m + 8) - 8 (3 m + 8)) = 0$

Step 4.1.3

Factor the polynomial by factoring out the greatest common factor, $3 m + 8$ .

$- 4 ((3 m + 8) (m - 8)) = 0$

Step 4.2

Remove unnecessary parentheses.

$- 4 (3 m + 8) (m - 8) = 0$