Algebra Examples

$(m + 1) (2 m + 7) = 3 (m + 4) - 9$

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 $(m + 1) (2 m + 7)$ .

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Step 1.1.1.1

Expand $(m + 1) (2 m + 7)$ using the FOIL Method.

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Step 1.1.1.1.1

Apply the distributive property.

$m (2 m + 7) + 1 (2 m + 7) = 3 (m + 4) - 9$

Step 1.1.1.1.2

Apply the distributive property.

$m (2 m) + m \cdot 7 + 1 (2 m + 7) = 3 (m + 4) - 9$

Step 1.1.1.1.3

Apply the distributive property.

$m (2 m) + m \cdot 7 + 1 (2 m) + 1 \cdot 7 = 3 (m + 4) - 9$

Step 1.1.1.2

Simplify and combine like terms.

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Step 1.1.1.2.1

Simplify each term.

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Step 1.1.1.2.1.1

Rewrite using the commutative property of multiplication.

$2 m \cdot m + m \cdot 7 + 1 (2 m) + 1 \cdot 7 = 3 (m + 4) - 9$

Step 1.1.1.2.1.2

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

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Step 1.1.1.2.1.2.1

Move $m$ .

$2 (m \cdot m) + m \cdot 7 + 1 (2 m) + 1 \cdot 7 = 3 (m + 4) - 9$

Step 1.1.1.2.1.2.2

Multiply $m$ by $m$ .

$2 m^{2} + m \cdot 7 + 1 (2 m) + 1 \cdot 7 = 3 (m + 4) - 9$

Step 1.1.1.2.1.3

Move $7$ to the left of $m$ .

$2 m^{2} + 7 \cdot m + 1 (2 m) + 1 \cdot 7 = 3 (m + 4) - 9$

Step 1.1.1.2.1.4

Multiply $2 m$ by $1$ .

$2 m^{2} + 7 m + 2 m + 1 \cdot 7 = 3 (m + 4) - 9$

Step 1.1.1.2.1.5

Multiply $7$ by $1$ .

$2 m^{2} + 7 m + 2 m + 7 = 3 (m + 4) - 9$

Step 1.1.1.2.2

Add $7 m$ and $2 m$ .

$2 m^{2} + 9 m + 7 = 3 (m + 4) - 9$

Step 1.2

Simplify the right side.

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Step 1.2.1

Simplify $3 (m + 4) - 9$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Apply the distributive property.

$2 m^{2} + 9 m + 7 = 3 m + 3 \cdot 4 - 9$

Step 1.2.1.1.2

Multiply $3$ by $4$ .

$2 m^{2} + 9 m + 7 = 3 m + 12 - 9$

Step 1.2.1.2

Subtract $9$ from $12$ .

$2 m^{2} + 9 m + 7 = 3 m + 3$

Step 1.3

Move all the expressions to the left side of the equation.

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Step 1.3.1

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

$2 m^{2} + 9 m + 7 - 3 m = 3$

Step 1.3.2

Subtract $3$ from both sides of the equation.

$2 m^{2} + 9 m + 7 - 3 m - 3 = 0$

Step 1.4

Simplify $2 m^{2} + 9 m + 7 - 3 m - 3$ .

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Step 1.4.1

Subtract $3 m$ from $9 m$ .

$2 m^{2} + 6 m + 7 - 3 = 0$

Step 1.4.2

Subtract $3$ from $7$ .

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

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 $a = 2$ , $b = 6$ , and $c = 4$ into the quadratic formula and solve for $m$ .

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot 4$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$m = \frac{- 6 \pm \sqrt{36 - 8 \cdot 4}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $4$ .

$m = \frac{- 6 \pm \sqrt{36 - 32}}{2 \cdot 2}$

Step 4.1.3

Subtract $32$ from $36$ .

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

Step 4.1.4

Rewrite $4$ as $2^{2}$ .

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

Step 4.1.5

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

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

Step 4.2

Multiply $2$ by $2$ .

$m = \frac{- 6 \pm 2}{4}$

Step 4.3

Simplify $\frac{- 6 \pm 2}{4}$ .

$m = \frac{- 3 \pm 1}{2}$

Step 5

The final answer is the combination of both solutions.

$m = - 1, - 2$