Solve Using the Quadratic Formula (m+2)(2m-6)=5(m-1)-12

$(m + 2) (2 m - 6) = 5 (m - 1) - 12$

Step 1

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

Simplify the left side.

Simplify $(m + 2) (2 m - 6)$ .

Expand $(m + 2) (2 m - 6)$ using the FOIL Method.

Apply the distributive property.

$m (2 m - 6) + 2 (2 m - 6) = 5 (m - 1) - 12$

Apply the distributive property.

$m (2 m) + m \cdot - 6 + 2 (2 m - 6) = 5 (m - 1) - 12$

Apply the distributive property.

$m (2 m) + m \cdot - 6 + 2 (2 m) + 2 \cdot - 6 = 5 (m - 1) - 12$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 m \cdot m + m \cdot - 6 + 2 (2 m) + 2 \cdot - 6 = 5 (m - 1) - 12$

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

Move $m$ .

$2 (m \cdot m) + m \cdot - 6 + 2 (2 m) + 2 \cdot - 6 = 5 (m - 1) - 12$

Multiply $m$ by $m$ .

$2 m^{2} + m \cdot - 6 + 2 (2 m) + 2 \cdot - 6 = 5 (m - 1) - 12$

Move $- 6$ to the left of $m$ .

$2 m^{2} - 6 \cdot m + 2 (2 m) + 2 \cdot - 6 = 5 (m - 1) - 12$

Multiply $2$ by $2$ .

$2 m^{2} - 6 m + 4 m + 2 \cdot - 6 = 5 (m - 1) - 12$

Multiply $2$ by $- 6$ .

$2 m^{2} - 6 m + 4 m - 12 = 5 (m - 1) - 12$

Add $- 6 m$ and $4 m$ .

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

Simplify the right side.

Simplify $5 (m - 1) - 12$ .

Simplify each term.

Apply the distributive property.

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

Multiply $5$ by $- 1$ .

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

Subtract $12$ from $- 5$ .

$2 m^{2} - 2 m - 12 = 5 m - 17$

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

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

$2 m^{2} - 2 m - 12 - 5 m = - 17$

Add $17$ to both sides of the equation.

$2 m^{2} - 2 m - 12 - 5 m + 17 = 0$

Simplify $2 m^{2} - 2 m - 12 - 5 m + 17$ .

Subtract $5 m$ from $- 2 m$ .

$2 m^{2} - 7 m - 12 + 17 = 0$

Add $- 12$ and $17$ .

$2 m^{2} - 7 m + 5 = 0$

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 $a = 2$ , $b = - 7$ , and $c = 5$ into the quadratic formula and solve for $m$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (2 \cdot 5)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 7$ to the power of $2$ .

$m = \frac{7 \pm \sqrt{49 - 4 \cdot 2 \cdot 5}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 5$ .

Multiply $- 4$ by $2$ .

$m = \frac{7 \pm \sqrt{49 - 8 \cdot 5}}{2 \cdot 2}$

Multiply $- 8$ by $5$ .

$m = \frac{7 \pm \sqrt{49 - 40}}{2 \cdot 2}$

Subtract $40$ from $49$ .

$m = \frac{7 \pm \sqrt{9}}{2 \cdot 2}$

Rewrite $9$ as $3^{2}$ .

$m = \frac{7 \pm \sqrt{3^{2}}}{2 \cdot 2}$

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

$m = \frac{7 \pm 3}{2 \cdot 2}$

Multiply $2$ by $2$ .

$m = \frac{7 \pm 3}{4}$

Step 5

The final answer is the combination of both solutions.

$m = \frac{5}{2}, 1$

Solve Using the Quadratic Formula (m+2)(2m-6)=5(m-1)-12

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