Solve for w 9w^2=30w-25

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

Subtract $30 w$ from both sides of the equation.

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

Add $25$ to both sides of the equation.

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

Factor using the perfect square rule.

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

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

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

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

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

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

Rewrite the polynomial.

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

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

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

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

$3 w - 5 = 0$

Solve for $w$ .

Add $5$ to both sides of the equation.

$3 w = 5$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $w$ by $1$ .

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$w = 1.\overline{6}$

Mixed Number Form:

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