Finite Math Examples

${(2 m + 4)}^{2} + {(3 m)}^{2} = {(5 m)}^{2}$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Apply the product rule to $3 m$ .

${(2 m + 4)}^{2} + 3^{2} m^{2} = {(5 m)}^{2}$

Step 1.1.1.2

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

${(2 m + 4)}^{2} + 9 m^{2} = {(5 m)}^{2}$

Step 1.2

Simplify the right side.

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Step 1.2.1

Simplify ${(5 m)}^{2}$ .

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Step 1.2.1.1

Apply the product rule to $5 m$ .

${(2 m + 4)}^{2} + 9 m^{2} = 5^{2} m^{2}$

Step 1.2.1.2

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

${(2 m + 4)}^{2} + 9 m^{2} = 25 m^{2}$

Step 1.3

Subtract $25 m^{2}$ from both sides of the equation.

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

Step 1.4

Simplify ${(2 m + 4)}^{2} + 9 m^{2} - 25 m^{2}$ .

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Rewrite ${(2 m + 4)}^{2}$ as $(2 m + 4) (2 m + 4)$ .

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

Step 1.4.1.2

Expand $(2 m + 4) (2 m + 4)$ using the FOIL Method.

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Step 1.4.1.2.1

Apply the distributive property.

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

Step 1.4.1.2.2

Apply the distributive property.

$2 m (2 m) + 2 m \cdot 4 + 4 (2 m + 4) + 9 m^{2} - 25 m^{2} = 0$

Step 1.4.1.2.3

Apply the distributive property.

$2 m (2 m) + 2 m \cdot 4 + 4 (2 m) + 4 \cdot 4 + 9 m^{2} - 25 m^{2} = 0$

Step 1.4.1.3

Simplify and combine like terms.

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Step 1.4.1.3.1

Simplify each term.

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Step 1.4.1.3.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 m \cdot m + 2 m \cdot 4 + 4 (2 m) + 4 \cdot 4 + 9 m^{2} - 25 m^{2} = 0$

Step 1.4.1.3.1.2

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

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Step 1.4.1.3.1.2.1

Move $m$ .

$2 \cdot 2 (m \cdot m) + 2 m \cdot 4 + 4 (2 m) + 4 \cdot 4 + 9 m^{2} - 25 m^{2} = 0$

Step 1.4.1.3.1.2.2

Multiply $m$ by $m$ .

$2 \cdot 2 m^{2} + 2 m \cdot 4 + 4 (2 m) + 4 \cdot 4 + 9 m^{2} - 25 m^{2} = 0$

Step 1.4.1.3.1.3

Multiply $2$ by $2$ .

$4 m^{2} + 2 m \cdot 4 + 4 (2 m) + 4 \cdot 4 + 9 m^{2} - 25 m^{2} = 0$

Step 1.4.1.3.1.4

Multiply $4$ by $2$ .

$4 m^{2} + 8 m + 4 (2 m) + 4 \cdot 4 + 9 m^{2} - 25 m^{2} = 0$

Step 1.4.1.3.1.5

Multiply $2$ by $4$ .

$4 m^{2} + 8 m + 8 m + 4 \cdot 4 + 9 m^{2} - 25 m^{2} = 0$

Step 1.4.1.3.1.6

Multiply $4$ by $4$ .

$4 m^{2} + 8 m + 8 m + 16 + 9 m^{2} - 25 m^{2} = 0$

Step 1.4.1.3.2

Add $8 m$ and $8 m$ .

$4 m^{2} + 16 m + 16 + 9 m^{2} - 25 m^{2} = 0$

Step 1.4.2

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

$13 m^{2} + 16 m + 16 - 25 m^{2} = 0$

Step 1.4.3

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

$- 12 m^{2} + 16 m + 16 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = - 12$ , $b = 16$ , and $c = 16$ into the quadratic formula and solve for $m$ .

$\frac{- 16 \pm \sqrt{16^{2} - 4 \cdot (- 12 \cdot 16)}}{2 \cdot - 12}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $16$ to the power of $2$ .

$m = \frac{- 16 \pm \sqrt{256 - 4 \cdot - 12 \cdot 16}}{2 \cdot - 12}$

Step 4.1.2

Multiply $- 4 \cdot - 12 \cdot 16$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 12$ .

$m = \frac{- 16 \pm \sqrt{256 + 48 \cdot 16}}{2 \cdot - 12}$

Step 4.1.2.2

Multiply $48$ by $16$ .

$m = \frac{- 16 \pm \sqrt{256 + 768}}{2 \cdot - 12}$

Step 4.1.3

Add $256$ and $768$ .

$m = \frac{- 16 \pm \sqrt{1024}}{2 \cdot - 12}$

Step 4.1.4

Rewrite $1024$ as $32^{2}$ .

$m = \frac{- 16 \pm \sqrt{32^{2}}}{2 \cdot - 12}$

Step 4.1.5

Pull terms out from under the radical, assuming positive real numbers.

$m = \frac{- 16 \pm 32}{2 \cdot - 12}$

Step 4.2

Multiply $2$ by $- 12$ .

$m = \frac{- 16 \pm 32}{- 24}$

Step 4.3

Simplify $\frac{- 16 \pm 32}{- 24}$ .

$m = \frac{2 \pm 4}{3}$

Step 5

The final answer is the combination of both solutions.

$m = 2, - \frac{2}{3}$