Algebra Examples

$(2 m + 3) (4 m + 3) = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify $(2 m + 3) (4 m + 3)$ .

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Step 1.1.1

Expand $(2 m + 3) (4 m + 3)$ using the FOIL Method.

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Step 1.1.1.1

Apply the distributive property.

$2 m (4 m + 3) + 3 (4 m + 3) = 0$

Step 1.1.1.2

Apply the distributive property.

$2 m (4 m) + 2 m \cdot 3 + 3 (4 m + 3) = 0$

Step 1.1.1.3

Apply the distributive property.

$2 m (4 m) + 2 m \cdot 3 + 3 (4 m) + 3 \cdot 3 = 0$

Step 1.1.2

Simplify and combine like terms.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 4 m \cdot m + 2 m \cdot 3 + 3 (4 m) + 3 \cdot 3 = 0$

Step 1.1.2.1.2

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

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Step 1.1.2.1.2.1

Move $m$ .

$2 \cdot 4 (m \cdot m) + 2 m \cdot 3 + 3 (4 m) + 3 \cdot 3 = 0$

Step 1.1.2.1.2.2

Multiply $m$ by $m$ .

$2 \cdot 4 m^{2} + 2 m \cdot 3 + 3 (4 m) + 3 \cdot 3 = 0$

Step 1.1.2.1.3

Multiply $2$ by $4$ .

$8 m^{2} + 2 m \cdot 3 + 3 (4 m) + 3 \cdot 3 = 0$

Step 1.1.2.1.4

Multiply $3$ by $2$ .

$8 m^{2} + 6 m + 3 (4 m) + 3 \cdot 3 = 0$

Step 1.1.2.1.5

Multiply $4$ by $3$ .

$8 m^{2} + 6 m + 12 m + 3 \cdot 3 = 0$

Step 1.1.2.1.6

Multiply $3$ by $3$ .

$8 m^{2} + 6 m + 12 m + 9 = 0$

Step 1.1.2.2

Add $6 m$ and $12 m$ .

$8 m^{2} + 18 m + 9 = 0$

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 $a = 8$ , $b = 18$ , and $c = 9$ into the quadratic formula and solve for $m$ .

$\frac{- 18 \pm \sqrt{18^{2} - 4 \cdot (8 \cdot 9)}}{2 \cdot 8}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $18$ to the power of $2$ .

$m = \frac{- 18 \pm \sqrt{324 - 4 \cdot 8 \cdot 9}}{2 \cdot 8}$

Step 4.1.2

Multiply $- 4 \cdot 8 \cdot 9$ .

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Step 4.1.2.1

Multiply $- 4$ by $8$ .

$m = \frac{- 18 \pm \sqrt{324 - 32 \cdot 9}}{2 \cdot 8}$

Step 4.1.2.2

Multiply $- 32$ by $9$ .

$m = \frac{- 18 \pm \sqrt{324 - 288}}{2 \cdot 8}$

Step 4.1.3

Subtract $288$ from $324$ .

$m = \frac{- 18 \pm \sqrt{36}}{2 \cdot 8}$

Step 4.1.4

Rewrite $36$ as $6^{2}$ .

$m = \frac{- 18 \pm \sqrt{6^{2}}}{2 \cdot 8}$

Step 4.1.5

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

$m = \frac{- 18 \pm 6}{2 \cdot 8}$

Step 4.2

Multiply $2$ by $8$ .

$m = \frac{- 18 \pm 6}{16}$

Step 4.3

Simplify $\frac{- 18 \pm 6}{16}$ .

$m = \frac{- 9 \pm 3}{8}$

Step 5

The final answer is the combination of both solutions.

$m = - \frac{3}{4}, - \frac{3}{2}$