Algebra Examples

$4 w^{2} + 5 = - 12 w$

Step 1

Add $12 w$ to both sides of the equation.

$4 w^{2} + 5 + 12 w = 0$

Step 2

Factor by grouping.

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Step 2.1

Reorder terms.

$4 w^{2} + 12 w + 5 = 0$

Step 2.2

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 $a \cdot c = 4 \cdot 5 = 20$ and whose sum is $b = 12$ .

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Step 2.2.1

Factor $12$ out of $12 w$ .

$4 w^{2} + 12 (w) + 5 = 0$

Step 2.2.2

Rewrite $12$ as $2$ plus $10$

$4 w^{2} + (2 + 10) w + 5 = 0$

Step 2.2.3

Apply the distributive property.

$4 w^{2} + 2 w + 10 w + 5 = 0$

Step 2.3

Factor out the greatest common factor from each group.

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Step 2.3.1

Group the first two terms and the last two terms.

$(4 w^{2} + 2 w) + 10 w + 5 = 0$

Step 2.3.2

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

$2 w (2 w + 1) + 5 (2 w + 1) = 0$

Step 2.4

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

$(2 w + 1) (2 w + 5) = 0$

Step 3

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

$2 w + 1 = 0$

$2 w + 5 = 0$

Step 4

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

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Step 4.1

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

$2 w + 1 = 0$

Step 4.2

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

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$2 w = - 1$

Step 4.2.2

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

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Step 4.2.2.1

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

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

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.

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

Step 4.2.2.2.1.2

Divide $w$ by $1$ .

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

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.

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

Step 5

Set $2 w + 5$ equal to $0$ and solve for $w$ .

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Step 5.1

Set $2 w + 5$ equal to $0$ .

$2 w + 5 = 0$

Step 5.2

Solve $2 w + 5 = 0$ for $w$ .

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Step 5.2.1

Subtract $5$ from both sides of the equation.

$2 w = - 5$

Step 5.2.2

Divide each term in $2 w = - 5$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 w = - 5$ by $2$ .

$\frac{2 w}{2} = \frac{- 5}{2}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{2 w}{2} = \frac{- 5}{2}$

Step 5.2.2.2.1.2

Divide $w$ by $1$ .

$w = \frac{- 5}{2}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

$w = - \frac{5}{2}$

Step 6

The final solution is all the values that make $(2 w + 1) (2 w + 5) = 0$ true.

$w = - \frac{1}{2}, - \frac{5}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$w = - \frac{1}{2}, - \frac{5}{2}$

Decimal Form:

$w = - 0.5, - 2.5$

Mixed Number Form:

$w = - \frac{1}{2}, - 2 \frac{1}{2}$