Precalculus Examples

${(2 m + 12)}^{2} + {(3 m)}^{2} = 5 m^{2}$

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $3 m$ .

${(2 m + 12)}^{2} + 3^{2} m^{2} = 5 m^{2}$

Step 1.2

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

${(2 m + 12)}^{2} + 9 m^{2} = 5 m^{2}$

Step 2

Move all terms containing $m$ to the left side of the equation.

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Step 2.1

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

${(2 m + 12)}^{2} + 9 m^{2} - 5 m^{2} = 0$

Step 2.2

Simplify each term.

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Step 2.2.1

Rewrite ${(2 m + 12)}^{2}$ as $(2 m + 12) (2 m + 12)$ .

$(2 m + 12) (2 m + 12) + 9 m^{2} - 5 m^{2} = 0$

Step 2.2.2

Expand $(2 m + 12) (2 m + 12)$ using the FOIL Method.

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Step 2.2.2.1

Apply the distributive property.

$2 m (2 m + 12) + 12 (2 m + 12) + 9 m^{2} - 5 m^{2} = 0$

Step 2.2.2.2

Apply the distributive property.

$2 m (2 m) + 2 m \cdot 12 + 12 (2 m + 12) + 9 m^{2} - 5 m^{2} = 0$

Step 2.2.2.3

Apply the distributive property.

$2 m (2 m) + 2 m \cdot 12 + 12 (2 m) + 12 \cdot 12 + 9 m^{2} - 5 m^{2} = 0$

Step 2.2.3

Simplify and combine like terms.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 m \cdot m + 2 m \cdot 12 + 12 (2 m) + 12 \cdot 12 + 9 m^{2} - 5 m^{2} = 0$

Step 2.2.3.1.2

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

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Step 2.2.3.1.2.1

Move $m$ .

$2 \cdot 2 (m \cdot m) + 2 m \cdot 12 + 12 (2 m) + 12 \cdot 12 + 9 m^{2} - 5 m^{2} = 0$

Step 2.2.3.1.2.2

Multiply $m$ by $m$ .

$2 \cdot 2 m^{2} + 2 m \cdot 12 + 12 (2 m) + 12 \cdot 12 + 9 m^{2} - 5 m^{2} = 0$

Step 2.2.3.1.3

Multiply $2$ by $2$ .

$4 m^{2} + 2 m \cdot 12 + 12 (2 m) + 12 \cdot 12 + 9 m^{2} - 5 m^{2} = 0$

Step 2.2.3.1.4

Multiply $12$ by $2$ .

$4 m^{2} + 24 m + 12 (2 m) + 12 \cdot 12 + 9 m^{2} - 5 m^{2} = 0$

Step 2.2.3.1.5

Multiply $2$ by $12$ .

$4 m^{2} + 24 m + 24 m + 12 \cdot 12 + 9 m^{2} - 5 m^{2} = 0$

Step 2.2.3.1.6

Multiply $12$ by $12$ .

$4 m^{2} + 24 m + 24 m + 144 + 9 m^{2} - 5 m^{2} = 0$

Step 2.2.3.2

Add $24 m$ and $24 m$ .

$4 m^{2} + 48 m + 144 + 9 m^{2} - 5 m^{2} = 0$

Step 2.3

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

$13 m^{2} + 48 m + 144 - 5 m^{2} = 0$

Step 2.4

Subtract $5 m^{2}$ from $13 m^{2}$ .

$8 m^{2} + 48 m + 144 = 0$

Step 3

Factor $8$ out of $8 m^{2} + 48 m + 144$ .

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Step 3.1

Factor $8$ out of $8 m^{2}$ .

$8 (m^{2}) + 48 m + 144 = 0$

Step 3.2

Factor $8$ out of $48 m$ .

$8 (m^{2}) + 8 (6 m) + 144 = 0$

Step 3.3

Factor $8$ out of $144$ .

$8 m^{2} + 8 (6 m) + 8 \cdot 18 = 0$

Step 3.4

Factor $8$ out of $8 m^{2} + 8 (6 m)$ .

$8 (m^{2} + 6 m) + 8 \cdot 18 = 0$

Step 3.5

Factor $8$ out of $8 (m^{2} + 6 m) + 8 \cdot 18$ .

$8 (m^{2} + 6 m + 18) = 0$

Step 4

Divide each term in $8 (m^{2} + 6 m + 18) = 0$ by $8$ and simplify.

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Step 4.1

Divide each term in $8 (m^{2} + 6 m + 18) = 0$ by $8$ .

$\frac{8 (m^{2} + 6 m + 18)}{8} = \frac{0}{8}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $8$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{8 (m^{2} + 6 m + 18)}{8} = \frac{0}{8}$

Step 4.2.1.2

Divide $m^{2} + 6 m + 18$ by $1$ .

$m^{2} + 6 m + 18 = \frac{0}{8}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $8$ .

$m^{2} + 6 m + 18 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 1$ , $b = 6$ , and $c = 18$ into the quadratic formula and solve for $m$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot 18)}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $6$ to the power of $2$ .

$m = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 18}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 18$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$m = \frac{- 6 \pm \sqrt{36 - 4 \cdot 18}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $18$ .

$m = \frac{- 6 \pm \sqrt{36 - 72}}{2 \cdot 1}$

Step 7.1.3

Subtract $72$ from $36$ .

$m = \frac{- 6 \pm \sqrt{- 36}}{2 \cdot 1}$

Step 7.1.4

Rewrite $- 36$ as $- 1 (36)$ .

$m = \frac{- 6 \pm \sqrt{- 1 \cdot 36}}{2 \cdot 1}$

Step 7.1.5

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$m = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{36}}{2 \cdot 1}$

Step 7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$m = \frac{- 6 \pm i \cdot \sqrt{36}}{2 \cdot 1}$

Step 7.1.7

Rewrite $36$ as $6^{2}$ .

$m = \frac{- 6 \pm i \cdot \sqrt{6^{2}}}{2 \cdot 1}$

Step 7.1.8

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

$m = \frac{- 6 \pm i \cdot 6}{2 \cdot 1}$

Step 7.1.9

Move $6$ to the left of $i$ .

$m = \frac{- 6 \pm 6 i}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$m = \frac{- 6 \pm 6 i}{2}$

Step 7.3

Simplify $\frac{- 6 \pm 6 i}{2}$ .

$m = - 3 \pm 3 i$

Step 8

The final answer is the combination of both solutions.

$m = - 3 + 3 i, - 3 - 3 i$