Solve by Factoring 30(p^2-1)=11p

$30 (p^{2} - 1) = 11 p$

Step 1

Subtract $11 p$ from both sides of the equation.

$30 (p^{2} - 1) - 11 p = 0$

Step 2

Simplify each term.

Apply the distributive property.

$30 p^{2} + 30 \cdot - 1 - 11 p = 0$

Multiply $30$ by $- 1$ .

$30 p^{2} - 30 - 11 p = 0$

Step 3

Factor by grouping.

Reorder terms.

$30 p^{2} - 11 p - 30 = 0$

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 $a \cdot c = 30 \cdot - 30 = - 900$ and whose sum is $b = - 11$ .

Factor $- 11$ out of $- 11 p$ .

$30 p^{2} - 11 p - 30 = 0$

Rewrite $- 11$ as $25$ plus $- 36$

$30 p^{2} + (25 - 36) p - 30 = 0$

Apply the distributive property.

$30 p^{2} + 25 p - 36 p - 30 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(30 p^{2} + 25 p) - 36 p - 30 = 0$

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

$5 p (6 p + 5) - 6 (6 p + 5) = 0$

Factor the polynomial by factoring out the greatest common factor, $6 p + 5$ .

$(6 p + 5) (5 p - 6) = 0$

Step 4

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

$6 p + 5 = 0$

$5 p - 6 = 0$

Step 5

Set $6 p + 5$ equal to $0$ and solve for $p$ .

Set $6 p + 5$ equal to $0$ .

$6 p + 5 = 0$

Solve $6 p + 5 = 0$ for $p$ .

Subtract $5$ from both sides of the equation.

$6 p = - 5$

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

Divide each term in $6 p = - 5$ by $6$ .

$\frac{6 p}{6} = \frac{- 5}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 p}{6} = \frac{- 5}{6}$

Divide $p$ by $1$ .

$p = \frac{- 5}{6}$

Simplify the right side.

Move the negative in front of the fraction.

$p = - \frac{5}{6}$

Step 6

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

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

$5 p - 6 = 0$

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

Add $6$ to both sides of the equation.

$5 p = 6$

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $p$ by $1$ .

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

Step 7

The final solution is all the values that make $(6 p + 5) (5 p - 6) = 0$ true.

$p = - \frac{5}{6}, \frac{6}{5}$

Solve by Factoring 30(p^2-1)=11p

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