Solve Using the Quadratic Formula (m+3)(5m+1)=3(m-4)+7

$(m + 3) (5 m + 1) = 3 (m - 4) + 7$

Step 1

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

Simplify the left side.

Simplify $(m + 3) (5 m + 1)$ .

Expand $(m + 3) (5 m + 1)$ using the FOIL Method.

Apply the distributive property.

$m (5 m + 1) + 3 (5 m + 1) = 3 (m - 4) + 7$

Apply the distributive property.

$m (5 m) + m \cdot 1 + 3 (5 m + 1) = 3 (m - 4) + 7$

Apply the distributive property.

$m (5 m) + m \cdot 1 + 3 (5 m) + 3 \cdot 1 = 3 (m - 4) + 7$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$5 m \cdot m + m \cdot 1 + 3 (5 m) + 3 \cdot 1 = 3 (m - 4) + 7$

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

Move $m$ .

$5 (m \cdot m) + m \cdot 1 + 3 (5 m) + 3 \cdot 1 = 3 (m - 4) + 7$

Multiply $m$ by $m$ .

$5 m^{2} + m \cdot 1 + 3 (5 m) + 3 \cdot 1 = 3 (m - 4) + 7$

Multiply $m$ by $1$ .

$5 m^{2} + m + 3 (5 m) + 3 \cdot 1 = 3 (m - 4) + 7$

Multiply $5$ by $3$ .

$5 m^{2} + m + 15 m + 3 \cdot 1 = 3 (m - 4) + 7$

Multiply $3$ by $1$ .

$5 m^{2} + m + 15 m + 3 = 3 (m - 4) + 7$

Add $m$ and $15 m$ .

$5 m^{2} + 16 m + 3 = 3 (m - 4) + 7$

Simplify the right side.

Simplify $3 (m - 4) + 7$ .

Simplify each term.

Apply the distributive property.

$5 m^{2} + 16 m + 3 = 3 m + 3 \cdot - 4 + 7$

Multiply $3$ by $- 4$ .

$5 m^{2} + 16 m + 3 = 3 m - 12 + 7$

Add $- 12$ and $7$ .

$5 m^{2} + 16 m + 3 = 3 m - 5$

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

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

$5 m^{2} + 16 m + 3 - 3 m = - 5$

Add $5$ to both sides of the equation.

$5 m^{2} + 16 m + 3 - 3 m + 5 = 0$

Simplify $5 m^{2} + 16 m + 3 - 3 m + 5$ .

Subtract $3 m$ from $16 m$ .

$5 m^{2} + 13 m + 3 + 5 = 0$

Add $3$ and $5$ .

$5 m^{2} + 13 m + 8 = 0$

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 $a = 5$ , $b = 13$ , and $c = 8$ into the quadratic formula and solve for $m$ .

$\frac{- 13 \pm \sqrt{13^{2} - 4 \cdot (5 \cdot 8)}}{2 \cdot 5}$

Step 4

Simplify.

Simplify the numerator.

Raise $13$ to the power of $2$ .

$m = \frac{- 13 \pm \sqrt{169 - 4 \cdot 5 \cdot 8}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot 8$ .

Multiply $- 4$ by $5$ .

$m = \frac{- 13 \pm \sqrt{169 - 20 \cdot 8}}{2 \cdot 5}$

Multiply $- 20$ by $8$ .

$m = \frac{- 13 \pm \sqrt{169 - 160}}{2 \cdot 5}$

Subtract $160$ from $169$ .

$m = \frac{- 13 \pm \sqrt{9}}{2 \cdot 5}$

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

$m = \frac{- 13 \pm \sqrt{3^{2}}}{2 \cdot 5}$

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

$m = \frac{- 13 \pm 3}{2 \cdot 5}$

Multiply $2$ by $5$ .

$m = \frac{- 13 \pm 3}{10}$

Step 5

The final answer is the combination of both solutions.

$m = - 1, - \frac{8}{5}$

Solve Using the Quadratic Formula (m+3)(5m+1)=3(m-4)+7

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