Pre-Algebra Examples

$9 w^{2} + 25 = - 30 w$

Step 1

Add $30 w$ to both sides of the equation.

$9 w^{2} + 25 + 30 w = 0$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rearrange terms.

$9 w^{2} + 30 w + 25 = 0$

Step 2.2

Rewrite $9 w^{2}$ as ${(3 w)}^{2}$ .

${(3 w)}^{2} + 30 w + 25 = 0$

Step 2.3

Rewrite $25$ as $5^{2}$ .

${(3 w)}^{2} + 30 w + 5^{2} = 0$

Step 2.4

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$30 w = 2 \cdot (3 w) \cdot 5$

Step 2.5

Rewrite the polynomial.

${(3 w)}^{2} + 2 \cdot (3 w) \cdot 5 + 5^{2} = 0$

Step 2.6

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 3 w$ and $b = 5$ .

${(3 w + 5)}^{2} = 0$

Step 3

Set the $3 w + 5$ equal to $0$ .

$3 w + 5 = 0$

Step 4

Solve for $w$ .

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Step 4.1

Subtract $5$ from both sides of the equation.

$3 w = - 5$

Step 4.2

Divide each term in $3 w = - 5$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 w = - 5$ by $3$ .

$\frac{3 w}{3} = \frac{- 5}{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{3 w}{3} = \frac{- 5}{3}$

Step 4.2.2.1.2

Divide $w$ by $1$ .

$w = \frac{- 5}{3}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move the negative in front of the fraction.

$w = - \frac{5}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$w = - \frac{5}{3}$

Decimal Form:

$w = - 1.\overline{6}$

Mixed Number Form:

$w = - 1 \frac{2}{3}$