Solve Using the Quadratic Formula (m-1)(3m+7)=3(m+5)-12

$(m - 1) (3 m + 7) = 3 (m + 5) - 12$

Step 1

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

Simplify the left side.

Simplify $(m - 1) (3 m + 7)$ .

Expand $(m - 1) (3 m + 7)$ using the FOIL Method.

Apply the distributive property.

$m (3 m + 7) - 1 (3 m + 7) = 3 (m + 5) - 12$

Apply the distributive property.

$m (3 m) + m \cdot 7 - 1 (3 m + 7) = 3 (m + 5) - 12$

Apply the distributive property.

$m (3 m) + m \cdot 7 - 1 (3 m) - 1 \cdot 7 = 3 (m + 5) - 12$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 m \cdot m + m \cdot 7 - 1 (3 m) - 1 \cdot 7 = 3 (m + 5) - 12$

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

Move $m$ .

$3 (m \cdot m) + m \cdot 7 - 1 (3 m) - 1 \cdot 7 = 3 (m + 5) - 12$

Multiply $m$ by $m$ .

$3 m^{2} + m \cdot 7 - 1 (3 m) - 1 \cdot 7 = 3 (m + 5) - 12$

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

$3 m^{2} + 7 \cdot m - 1 (3 m) - 1 \cdot 7 = 3 (m + 5) - 12$

Multiply $3$ by $- 1$ .

$3 m^{2} + 7 m - 3 m - 1 \cdot 7 = 3 (m + 5) - 12$

Multiply $- 1$ by $7$ .

$3 m^{2} + 7 m - 3 m - 7 = 3 (m + 5) - 12$

Subtract $3 m$ from $7 m$ .

$3 m^{2} + 4 m - 7 = 3 (m + 5) - 12$

Simplify the right side.

Simplify $3 (m + 5) - 12$ .

Simplify each term.

Apply the distributive property.

$3 m^{2} + 4 m - 7 = 3 m + 3 \cdot 5 - 12$

Multiply $3$ by $5$ .

$3 m^{2} + 4 m - 7 = 3 m + 15 - 12$

Subtract $12$ from $15$ .

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

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

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

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

Subtract $3$ from both sides of the equation.

$3 m^{2} + 4 m - 7 - 3 m - 3 = 0$

Simplify $3 m^{2} + 4 m - 7 - 3 m - 3$ .

Subtract $3 m$ from $4 m$ .

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

Subtract $3$ from $- 7$ .

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

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 $a = 3$ , $b = 1$ , and $c = - 10$ into the quadratic formula and solve for $m$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (3 \cdot - 10)}}{2 \cdot 3}$

Step 4

Simplify.

Simplify the numerator.

One to any power is one.

$m = \frac{- 1 \pm \sqrt{1 - 4 \cdot 3 \cdot - 10}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 10$ .

Multiply $- 4$ by $3$ .

$m = \frac{- 1 \pm \sqrt{1 - 12 \cdot - 10}}{2 \cdot 3}$

Multiply $- 12$ by $- 10$ .

$m = \frac{- 1 \pm \sqrt{1 + 120}}{2 \cdot 3}$

Add $1$ and $120$ .

$m = \frac{- 1 \pm \sqrt{121}}{2 \cdot 3}$

Rewrite $121$ as $11^{2}$ .

$m = \frac{- 1 \pm \sqrt{11^{2}}}{2 \cdot 3}$

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

$m = \frac{- 1 \pm 11}{2 \cdot 3}$

Multiply $2$ by $3$ .

$m = \frac{- 1 \pm 11}{6}$

Step 5

The final answer is the combination of both solutions.

$m = \frac{5}{3}, - 2$

Solve Using the Quadratic Formula (m-1)(3m+7)=3(m+5)-12

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