Solve by Factoring 6(p^2-1)=5p

$6 (p^{2} - 1) = 5 p$

Step 1

Subtract $5 p$ from both sides of the equation.

$6 (p^{2} - 1) - 5 p = 0$

Step 2

Simplify each term.

Apply the distributive property.

$6 p^{2} + 6 \cdot - 1 - 5 p = 0$

Multiply $6$ by $- 1$ .

$6 p^{2} - 6 - 5 p = 0$

Step 3

Factor by grouping.

Reorder terms.

$6 p^{2} - 5 p - 6 = 0$

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 $b = - 5$ .

Factor $- 5$ out of $- 5 p$ .

$6 p^{2} - 5 p - 6 = 0$

Rewrite $- 5$ as $4$ plus $- 9$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$2 p (3 p + 2) - 3 (3 p + 2) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 p + 2$ .

$(3 p + 2) (2 p - 3) = 0$

Step 4

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

$3 p + 2 = 0$

$2 p - 3 = 0$

Step 5

Set $3 p + 2$ equal to $0$ and solve for $p$ .

Set $3 p + 2$ equal to $0$ .

$3 p + 2 = 0$

Solve $3 p + 2 = 0$ for $p$ .

Subtract $2$ from both sides of the equation.

$3 p = - 2$

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

Divide each term in $3 p = - 2$ by $3$ .

$\frac{3 p}{3} = \frac{- 2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 p}{3} = \frac{- 2}{3}$

Divide $p$ by $1$ .

$p = \frac{- 2}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$p = - \frac{2}{3}$

Step 6

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

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

$2 p - 3 = 0$

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

Add $3$ to both sides of the equation.

$2 p = 3$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $p$ by $1$ .

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

Step 7

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

$p = - \frac{2}{3}, \frac{3}{2}$

Solve by Factoring 6(p^2-1)=5p

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