Basic Math Examples

$3 w^{2} - 3 w + 6 + (- 7 w + 9) - (5 w^{2} - 2)$

Step 1

Remove parentheses.

$3 w^{2} - 3 w + 6 - 7 w + 9 - (5 w^{2} - 2)$

Step 2

Regroup terms.

$3 w^{2} - 3 w - (5 w^{2} - 2) + 6 - 7 w + 9$

Step 3

Apply the distributive property.

$3 w^{2} - 3 w - (5 w^{2}) - - 2 + 6 - 7 w + 9$

Step 4

Multiply $5$ by $- 1$ .

$3 w^{2} - 3 w - 5 w^{2} - - 2 + 6 - 7 w + 9$

Step 5

Multiply $- 1$ by $- 2$ .

$3 w^{2} - 3 w - 5 w^{2} + 2 + 6 - 7 w + 9$

Step 6

Subtract $5 w^{2}$ from $3 w^{2}$ .

$- 2 w^{2} - 3 w + 2 + 6 - 7 w + 9$

Step 7

Rewrite $- 2 w^{2} - 3 w + 2$ in a factored form.

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Step 7.1

Factor $- 1$ out of $- 2 w^{2} - 3 w + 2$ .

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Step 7.1.1

Factor $- 1$ out of $- 2 w^{2}$ .

$- (2 w^{2}) - 3 w + 2 + 6 - 7 w + 9$

Step 7.1.2

Factor $- 1$ out of $- 3 w$ .

$- (2 w^{2}) - (3 w) + 2 + 6 - 7 w + 9$

Step 7.1.3

Rewrite $2$ as $- 1 (- 2)$ .

$- (2 w^{2}) - (3 w) - 1 (- 2) + 6 - 7 w + 9$

Step 7.1.4

Factor $- 1$ out of $- (2 w^{2}) - (3 w)$ .

$- (2 w^{2} + 3 w) - 1 (- 2) + 6 - 7 w + 9$

Step 7.1.5

Factor $- 1$ out of $- (2 w^{2} + 3 w) - 1 (- 2)$ .

$- (2 w^{2} + 3 w - 2) + 6 - 7 w + 9$

Step 7.2

Factor.

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Step 7.2.1

Factor by grouping.

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Step 7.2.1.1

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 $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = 3$ .

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Step 7.2.1.1.1

Factor $3$ out of $3 w$ .

$- (2 w^{2} + 3 (w) - 2) + 6 - 7 w + 9$

Step 7.2.1.1.2

Rewrite $3$ as $- 1$ plus $4$

$- (2 w^{2} + (- 1 + 4) w - 2) + 6 - 7 w + 9$

Step 7.2.1.1.3

Apply the distributive property.

$- (2 w^{2} - 1 w + 4 w - 2) + 6 - 7 w + 9$

Step 7.2.1.2

Factor out the greatest common factor from each group.

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Step 7.2.1.2.1

Group the first two terms and the last two terms.

$- ((2 w^{2} - 1 w) + 4 w - 2) + 6 - 7 w + 9$

Step 7.2.1.2.2

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

$- (w (2 w - 1) + 2 (2 w - 1)) + 6 - 7 w + 9$

Step 7.2.1.3

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

$- ((2 w - 1) (w + 2)) + 6 - 7 w + 9$

Step 7.2.2

Remove unnecessary parentheses.

$- (2 w - 1) (w + 2) + 6 - 7 w + 9$

Step 8

Add $6$ and $9$ .

$- (2 w - 1) (w + 2) - 7 w + 15$

Step 9

Factor $- 1$ out of $- 7 w + 15$ .

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Step 9.1

Factor $- 1$ out of $- 7 w$ .

$- (2 w - 1) (w + 2) - (7 w) + 15$

Step 9.2

Rewrite $15$ as $- 1 (- 15)$ .

$- (2 w - 1) (w + 2) - (7 w) - 1 (- 15)$

Step 9.3

Factor $- 1$ out of $- (7 w) - 1 (- 15)$ .

$- (2 w - 1) (w + 2) - (7 w - 15)$

Step 10

Factor $- 1$ out of $- (2 w - 1) (w + 2) - (7 w - 15)$ .

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Step 10.1

Factor $- 1$ out of $- (2 w - 1) (w + 2)$ .

$- ((2 w - 1) (w + 2)) - (7 w - 15)$

Step 10.2

Factor $- 1$ out of $- ((2 w - 1) (w + 2)) - (7 w - 15)$ .

$- ((2 w - 1) (w + 2) + 7 w - 15)$

Step 11

Expand $(2 w - 1) (w + 2)$ using the FOIL Method.

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Step 11.1

Apply the distributive property.

$- (2 w (w + 2) - 1 (w + 2) + 7 w - 15)$

Step 11.2

Apply the distributive property.

$- (2 w \cdot w + 2 w \cdot 2 - 1 (w + 2) + 7 w - 15)$

Step 11.3

Apply the distributive property.

$- (2 w \cdot w + 2 w \cdot 2 - 1 w - 1 \cdot 2 + 7 w - 15)$

Step 12

Simplify and combine like terms.

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Step 12.1

Simplify each term.

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Step 12.1.1

Multiply $w$ by $w$ by adding the exponents.

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Step 12.1.1.1

Move $w$ .

$- (2 (w \cdot w) + 2 w \cdot 2 - 1 w - 1 \cdot 2 + 7 w - 15)$

Step 12.1.1.2

Multiply $w$ by $w$ .

$- (2 w^{2} + 2 w \cdot 2 - 1 w - 1 \cdot 2 + 7 w - 15)$

Step 12.1.2

Multiply $2$ by $2$ .

$- (2 w^{2} + 4 w - 1 w - 1 \cdot 2 + 7 w - 15)$

Step 12.1.3

Rewrite $- 1 w$ as $- w$ .

$- (2 w^{2} + 4 w - w - 1 \cdot 2 + 7 w - 15)$

Step 12.1.4

Multiply $- 1$ by $2$ .

$- (2 w^{2} + 4 w - w - 2 + 7 w - 15)$

Step 12.2

Subtract $w$ from $4 w$ .

$- (2 w^{2} + 3 w - 2 + 7 w - 15)$

Step 13

Add $3 w$ and $7 w$ .

$- (2 w^{2} + 10 w - 2 - 15)$

Step 14

Subtract $15$ from $- 2$ .

$- (2 w^{2} + 10 w - 17)$