Algebra Examples

${(3 m - 2)}^{2} - m^{2}$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(3 m - 2)}^{2}$ as $(3 m - 2) (3 m - 2)$ .

$(3 m - 2) (3 m - 2) - m^{2}$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$3 m (3 m - 2) - 2 (3 m - 2) - m^{2}$

Step 1.2.2

Apply the distributive property.

$3 m (3 m) + 3 m \cdot - 2 - 2 (3 m - 2) - m^{2}$

Step 1.2.3

Apply the distributive property.

$3 m (3 m) + 3 m \cdot - 2 - 2 (3 m) - 2 \cdot - 2 - m^{2}$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$3 \cdot 3 m \cdot m + 3 m \cdot - 2 - 2 (3 m) - 2 \cdot - 2 - m^{2}$

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $m$ .

$3 \cdot 3 (m \cdot m) + 3 m \cdot - 2 - 2 (3 m) - 2 \cdot - 2 - m^{2}$

Step 1.3.1.2.2

Multiply $m$ by $m$ .

$3 \cdot 3 m^{2} + 3 m \cdot - 2 - 2 (3 m) - 2 \cdot - 2 - m^{2}$

Step 1.3.1.3

Multiply $3$ by $3$ .

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

Step 1.3.1.4

Multiply $- 2$ by $3$ .

$9 m^{2} - 6 m - 2 (3 m) - 2 \cdot - 2 - m^{2}$

Step 1.3.1.5

Multiply $3$ by $- 2$ .

$9 m^{2} - 6 m - 6 m - 2 \cdot - 2 - m^{2}$

Step 1.3.1.6

Multiply $- 2$ by $- 2$ .

$9 m^{2} - 6 m - 6 m + 4 - m^{2}$

Step 1.3.2

Subtract $6 m$ from $- 6 m$ .

$9 m^{2} - 12 m + 4 - m^{2}$

Step 2

Subtract $m^{2}$ from $9 m^{2}$ .

$8 m^{2} - 12 m + 4$

Step 3

Factor out the GCF of $4$ from $8 m^{2} - 12 m + 4$ .

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Step 3.1

Factor out the GCF of $4$ from each term in the polynomial.

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Step 3.1.1

Factor out the GCF of $4$ from the expression $8 m^{2}$ .

$4 (2 m^{2}) - 12 m + 4$

Step 3.1.2

Factor out the GCF of $4$ from the expression $- 12 m$ .

$4 (2 m^{2}) + 4 (- 3 m) + 4$

Step 3.1.3

Factor out the GCF of $4$ from the expression $4$ .

$4 (2 m^{2}) + 4 (- 3 m) + 4 (1)$

Step 3.2

Since all the terms share a common factor of $4$ , it can be factored out of each term.

$4 (2 m^{2} - 3 m + 1)$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = - 3$ .

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Step 4.1.1

Factor $- 3$ out of $- 3 m$ .

$4 (2 m^{2} - 3 m + 1)$

Step 4.1.2

Rewrite $- 3$ as $- 1$ plus $- 2$

$4 (2 m^{2} + (- 1 - 2) m + 1)$

Step 4.1.3

Apply the distributive property.

$4 (2 m^{2} - 1 m - 2 m + 1)$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$4 ((2 m^{2} - 1 m) - 2 m + 1)$

Step 4.2.2

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

$4 (m (2 m - 1) - (2 m - 1))$

Step 4.3

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

$4 ((2 m - 1) (m - 1))$