Algebra Examples

$(4 m + 2) (3 m - 9) = 0$

Step 1

Simplify $(4 m + 2) (3 m - 9)$ .

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Step 1.1

Expand $(4 m + 2) (3 m - 9)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$4 m (3 m - 9) + 2 (3 m - 9) = 0$

Step 1.1.2

Apply the distributive property.

$4 m (3 m) + 4 m \cdot - 9 + 2 (3 m - 9) = 0$

Step 1.1.3

Apply the distributive property.

$4 m (3 m) + 4 m \cdot - 9 + 2 (3 m) + 2 \cdot - 9 = 0$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite using the commutative property of multiplication.

$4 \cdot 3 m \cdot m + 4 m \cdot - 9 + 2 (3 m) + 2 \cdot - 9 = 0$

Step 1.2.1.2

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

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Step 1.2.1.2.1

Move $m$ .

$4 \cdot 3 (m \cdot m) + 4 m \cdot - 9 + 2 (3 m) + 2 \cdot - 9 = 0$

Step 1.2.1.2.2

Multiply $m$ by $m$ .

$4 \cdot 3 m^{2} + 4 m \cdot - 9 + 2 (3 m) + 2 \cdot - 9 = 0$

Step 1.2.1.3

Multiply $4$ by $3$ .

$12 m^{2} + 4 m \cdot - 9 + 2 (3 m) + 2 \cdot - 9 = 0$

Step 1.2.1.4

Multiply $- 9$ by $4$ .

$12 m^{2} - 36 m + 2 (3 m) + 2 \cdot - 9 = 0$

Step 1.2.1.5

Multiply $3$ by $2$ .

$12 m^{2} - 36 m + 6 m + 2 \cdot - 9 = 0$

Step 1.2.1.6

Multiply $2$ by $- 9$ .

$12 m^{2} - 36 m + 6 m - 18 = 0$

Step 1.2.2

Add $- 36 m$ and $6 m$ .

$12 m^{2} - 30 m - 18 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $6$ out of $12 m^{2} - 30 m - 18$ .

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Step 2.1.1

Factor $6$ out of $12 m^{2}$ .

$6 (2 m^{2}) - 30 m - 18 = 0$

Step 2.1.2

Factor $6$ out of $- 30 m$ .

$6 (2 m^{2}) + 6 (- 5 m) - 18 = 0$

Step 2.1.3

Factor $6$ out of $- 18$ .

$6 (2 m^{2}) + 6 (- 5 m) + 6 (- 3) = 0$

Step 2.1.4

Factor $6$ out of $6 (2 m^{2}) + 6 (- 5 m)$ .

$6 (2 m^{2} - 5 m) + 6 (- 3) = 0$

Step 2.1.5

Factor $6$ out of $6 (2 m^{2} - 5 m) + 6 (- 3)$ .

$6 (2 m^{2} - 5 m - 3) = 0$

Step 2.2

Factor.

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Step 2.2.1

Factor by grouping.

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Step 2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 5$ .

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Step 2.2.1.1.1

Factor $- 5$ out of $- 5 m$ .

$6 (2 m^{2} - 5 m - 3) = 0$

Step 2.2.1.1.2

Rewrite $- 5$ as $1$ plus $- 6$

$6 (2 m^{2} + (1 - 6) m - 3) = 0$

Step 2.2.1.1.3

Apply the distributive property.

$6 (2 m^{2} + 1 m - 6 m - 3) = 0$

Step 2.2.1.1.4

Multiply $m$ by $1$ .

$6 (2 m^{2} + m - 6 m - 3) = 0$

Step 2.2.1.2

Factor out the greatest common factor from each group.

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Step 2.2.1.2.1

Group the first two terms and the last two terms.

$6 ((2 m^{2} + m) - 6 m - 3) = 0$

Step 2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$6 (m (2 m + 1) - 3 (2 m + 1)) = 0$

Step 2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 m + 1$ .

$6 ((2 m + 1) (m - 3)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$6 (2 m + 1) (m - 3) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 m + 1 = 0$

$m - 3 = 0$

Step 4

Set $2 m + 1$ equal to $0$ and solve for $m$ .

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Step 4.1

Set $2 m + 1$ equal to $0$ .

$2 m + 1 = 0$

Step 4.2

Solve $2 m + 1 = 0$ for $m$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$2 m = - 1$

Step 4.2.2

Divide each term in $2 m = - 1$ by $2$ and simplify.

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Step 4.2.2.1

Divide each term in $2 m = - 1$ by $2$ .

$\frac{2 m}{2} = \frac{- 1}{2}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{2 m}{2} = \frac{- 1}{2}$

Step 4.2.2.2.1.2

Divide $m$ by $1$ .

$m = \frac{- 1}{2}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

$m = - \frac{1}{2}$

Step 5

Set $m - 3$ equal to $0$ and solve for $m$ .

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Step 5.1

Set $m - 3$ equal to $0$ .

$m - 3 = 0$

Step 5.2

Add $3$ to both sides of the equation.

$m = 3$

Step 6

The final solution is all the values that make $6 (2 m + 1) (m - 3) = 0$ true.

$m = - \frac{1}{2}, 3$

Step 7