Solve by Factoring 10b^2=27b-18

$10 b^{2} = 27 b - 18$

Step 1

Move all the expressions to the left side of the equation.

Subtract $27 b$ from both sides of the equation.

$10 b^{2} - 27 b = - 18$

Add $18$ to both sides of the equation.

$10 b^{2} - 27 b + 18 = 0$

Step 2

Factor by grouping.

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 $a \cdot c = 10 \cdot 18 = 180$ and whose sum is $b = - 27$ .

Factor $- 27$ out of $- 27 b$ .

$10 b^{2} - 27 b + 18 = 0$

Rewrite $- 27$ as $- 12$ plus $- 15$

$10 b^{2} + (- 12 - 15) b + 18 = 0$

Apply the distributive property.

$10 b^{2} - 12 b - 15 b + 18 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(10 b^{2} - 12 b) - 15 b + 18 = 0$

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

$2 b (5 b - 6) - 3 (5 b - 6) = 0$

Factor the polynomial by factoring out the greatest common factor, $5 b - 6$ .

$(5 b - 6) (2 b - 3) = 0$

Step 3

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

$5 b - 6 = 0$

$2 b - 3 = 0$

Step 4

Set $5 b - 6$ equal to $0$ and solve for $b$ .

Set $5 b - 6$ equal to $0$ .

$5 b - 6 = 0$

Solve $5 b - 6 = 0$ for $b$ .

Add $6$ to both sides of the equation.

$5 b = 6$

Divide each term in $5 b = 6$ by $5$ and simplify.

Divide each term in $5 b = 6$ by $5$ .

$\frac{5 b}{5} = \frac{6}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 b}{5} = \frac{6}{5}$

Divide $b$ by $1$ .

$b = \frac{6}{5}$

Step 5

Set $2 b - 3$ equal to $0$ and solve for $b$ .

Set $2 b - 3$ equal to $0$ .

$2 b - 3 = 0$

Solve $2 b - 3 = 0$ for $b$ .

Add $3$ to both sides of the equation.

$2 b = 3$

Divide each term in $2 b = 3$ by $2$ and simplify.

Divide each term in $2 b = 3$ by $2$ .

$\frac{2 b}{2} = \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 b}{2} = \frac{3}{2}$

Divide $b$ by $1$ .

$b = \frac{3}{2}$

Step 6

The final solution is all the values that make $(5 b - 6) (2 b - 3) = 0$ true.

$b = \frac{6}{5}, \frac{3}{2}$

Solve by Factoring 10b^2=27b-18

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