Solve by Factoring 6b^2-13b+3=-3

$6 b^{2} - 13 b + 3 = - 3$

Step 1

Add $3$ to both sides of the equation.

$6 b^{2} - 13 b + 3 + 3 = 0$

Step 2

Add $3$ and $3$ .

$6 b^{2} - 13 b + 6 = 0$

Step 3

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 $a \cdot c = 6 \cdot 6 = 36$ and whose sum is $b = - 13$ .

Factor $- 13$ out of $- 13 b$ .

$6 b^{2} - 13 b + 6 = 0$

Rewrite $- 13$ as $- 4$ plus $- 9$

$6 b^{2} + (- 4 - 9) b + 6 = 0$

Apply the distributive property.

$6 b^{2} - 4 b - 9 b + 6 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(6 b^{2} - 4 b) - 9 b + 6 = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $3 b - 2$ .

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

Step 4

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

$3 b - 2 = 0$

$2 b - 3 = 0$

Step 5

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

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

$3 b - 2 = 0$

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

Add $2$ to both sides of the equation.

$3 b = 2$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $b$ by $1$ .

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

Step 6

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 7

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

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

Solve by Factoring 6b^2-13b+3=-3

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