Algebra Examples

$(m - 3) (3 m + 4) = 5 (m + 1) + 8$

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

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Step 1.1.1.1

Expand $(m - 3) (3 m + 4)$ using the FOIL Method.

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Step 1.1.1.1.1

Apply the distributive property.

$m (3 m + 4) - 3 (3 m + 4) = 5 (m + 1) + 8$

Step 1.1.1.1.2

Apply the distributive property.

$m (3 m) + m \cdot 4 - 3 (3 m + 4) = 5 (m + 1) + 8$

Step 1.1.1.1.3

Apply the distributive property.

$m (3 m) + m \cdot 4 - 3 (3 m) - 3 \cdot 4 = 5 (m + 1) + 8$

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.

$3 m \cdot m + m \cdot 4 - 3 (3 m) - 3 \cdot 4 = 5 (m + 1) + 8$

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 $m$ .

$3 (m \cdot m) + m \cdot 4 - 3 (3 m) - 3 \cdot 4 = 5 (m + 1) + 8$

Step 1.1.1.2.1.2.2

Multiply $m$ by $m$ .

$3 m^{2} + m \cdot 4 - 3 (3 m) - 3 \cdot 4 = 5 (m + 1) + 8$

Step 1.1.1.2.1.3

Move $4$ to the left of $m$ .

$3 m^{2} + 4 \cdot m - 3 (3 m) - 3 \cdot 4 = 5 (m + 1) + 8$

Step 1.1.1.2.1.4

Multiply $3$ by $- 3$ .

$3 m^{2} + 4 m - 9 m - 3 \cdot 4 = 5 (m + 1) + 8$

Step 1.1.1.2.1.5

Multiply $- 3$ by $4$ .

$3 m^{2} + 4 m - 9 m - 12 = 5 (m + 1) + 8$

Step 1.1.1.2.2

Subtract $9 m$ from $4 m$ .

$3 m^{2} - 5 m - 12 = 5 (m + 1) + 8$

Step 1.2

Simplify the right side.

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Step 1.2.1

Simplify $5 (m + 1) + 8$ .

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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.

$3 m^{2} - 5 m - 12 = 5 m + 5 \cdot 1 + 8$

Step 1.2.1.1.2

Multiply $5$ by $1$ .

$3 m^{2} - 5 m - 12 = 5 m + 5 + 8$

Step 1.2.1.2

Add $5$ and $8$ .

$3 m^{2} - 5 m - 12 = 5 m + 13$

Step 1.3

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

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Step 1.3.1

Subtract $5 m$ from both sides of the equation.

$3 m^{2} - 5 m - 12 - 5 m = 13$

Step 1.3.2

Subtract $13$ from both sides of the equation.

$3 m^{2} - 5 m - 12 - 5 m - 13 = 0$

Step 1.4

Simplify $3 m^{2} - 5 m - 12 - 5 m - 13$ .

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Step 1.4.1

Subtract $5 m$ from $- 5 m$ .

$3 m^{2} - 10 m - 12 - 13 = 0$

Step 1.4.2

Subtract $13$ from $- 12$ .

$3 m^{2} - 10 m - 25 = 0$

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 $a = 3$ , $b = - 10$ , and $c = - 25$ into the quadratic formula and solve for $m$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (3 \cdot - 25)}}{2 \cdot 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 10$ to the power of $2$ .

$m = \frac{10 \pm \sqrt{100 - 4 \cdot 3 \cdot - 25}}{2 \cdot 3}$

Step 4.1.2

Multiply $- 4 \cdot 3 \cdot - 25$ .

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Step 4.1.2.1

Multiply $- 4$ by $3$ .

$m = \frac{10 \pm \sqrt{100 - 12 \cdot - 25}}{2 \cdot 3}$

Step 4.1.2.2

Multiply $- 12$ by $- 25$ .

$m = \frac{10 \pm \sqrt{100 + 300}}{2 \cdot 3}$

Step 4.1.3

Add $100$ and $300$ .

$m = \frac{10 \pm \sqrt{400}}{2 \cdot 3}$

Step 4.1.4

Rewrite $400$ as $20^{2}$ .

$m = \frac{10 \pm \sqrt{20^{2}}}{2 \cdot 3}$

Step 4.1.5

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

$m = \frac{10 \pm 20}{2 \cdot 3}$

Step 4.2

Multiply $2$ by $3$ .

$m = \frac{10 \pm 20}{6}$

Step 4.3

Simplify $\frac{10 \pm 20}{6}$ .

$m = \frac{5 \pm 10}{3}$

Step 5

The final answer is the combination of both solutions.

$m = 5, - \frac{5}{3}$