Pre-Algebra Examples

$(m + 2) (m + 3) = m + 2 (m - 2)$

Step 1

Simplify $(m + 2) (m + 3)$ .

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Step 1.1

Rewrite.

$0 + 0 + (m + 2) (m + 3) = m + 2 (m - 2)$

Step 1.2

Simplify by adding zeros.

$(m + 2) (m + 3) = m + 2 (m - 2)$

Step 1.3

Expand $(m + 2) (m + 3)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$m (m + 3) + 2 (m + 3) = m + 2 (m - 2)$

Step 1.3.2

Apply the distributive property.

$m \cdot m + m \cdot 3 + 2 (m + 3) = m + 2 (m - 2)$

Step 1.3.3

Apply the distributive property.

$m \cdot m + m \cdot 3 + 2 m + 2 \cdot 3 = m + 2 (m - 2)$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $m$ by $m$ .

$m^{2} + m \cdot 3 + 2 m + 2 \cdot 3 = m + 2 (m - 2)$

Step 1.4.1.2

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

$m^{2} + 3 \cdot m + 2 m + 2 \cdot 3 = m + 2 (m - 2)$

Step 1.4.1.3

Multiply $2$ by $3$ .

$m^{2} + 3 m + 2 m + 6 = m + 2 (m - 2)$

Step 1.4.2

Add $3 m$ and $2 m$ .

$m^{2} + 5 m + 6 = m + 2 (m - 2)$

Step 2

Simplify $m + 2 (m - 2)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$m^{2} + 5 m + 6 = m + 2 m + 2 \cdot - 2$

Step 2.1.2

Multiply $2$ by $- 2$ .

$m^{2} + 5 m + 6 = m + 2 m - 4$

Step 2.2

Add $m$ and $2 m$ .

$m^{2} + 5 m + 6 = 3 m - 4$

Step 3

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

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Step 3.1

Subtract $3 m$ from both sides of the equation.

$m^{2} + 5 m + 6 - 3 m = - 4$

Step 3.2

Subtract $3 m$ from $5 m$ .

$m^{2} + 2 m + 6 = - 4$

Step 4

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

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Step 4.1

Add $4$ to both sides of the equation.

$m^{2} + 2 m + 6 + 4 = 0$

Step 4.2

Add $6$ and $4$ .

$m^{2} + 2 m + 10 = 0$

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 $a = 1$ , $b = 2$ , and $c = 10$ into the quadratic formula and solve for $m$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot 10)}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$m = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 10$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$m = \frac{- 2 \pm \sqrt{4 - 4 \cdot 10}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $10$ .

$m = \frac{- 2 \pm \sqrt{4 - 40}}{2 \cdot 1}$

Step 7.1.3

Subtract $40$ from $4$ .

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

Step 7.1.4

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

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

Step 7.1.5

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

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

Step 7.1.6

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

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

Step 7.1.7

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

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

Step 7.1.8

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

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

Step 7.1.9

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

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

Step 7.2

Multiply $2$ by $1$ .

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

Step 7.3

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

$m = - 1 \pm 3 i$

Step 8

The final answer is the combination of both solutions.

$m = - 1 + 3 i, - 1 - 3 i$