Solve 7p^2=-8p-1

$$7 p ^ { 2 } = - 8 p - 1$$

Solve for p

$$p=-\frac{1}{7}\approx -0.142857143$$

$$p=-1$$

Steps Using Factoring By Grouping

Add 8p to both sides.

$$7p^{2}+8p=-1$$

Add 1 to both sides.

$$7p^{2}+8p+1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 7p^{2}+ap+bp+1. To find a and b, set up a system to be solved.

$$a+b=8 ab=7\times 1=7$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=7$$

Rewrite 7p^{2}+8p+1 as \left(7p^{2}+p\right)+\left(7p+1\right).

$$\left(7p^{2}+p\right)+\left(7p+1\right)$$

Factor out p in 7p^{2}+p.

$$p\left(7p+1\right)+7p+1$$

Factor out common term 7p+1 by using distributive property.

$$\left(7p+1\right)\left(p+1\right)$$

To find equation solutions, solve 7p+1=0 and p+1=0.

$$p=-\frac{1}{7} p=-1$$

Steps Using the Quadratic Formula

Add 8p to both sides.

$$7p^{2}+8p=-1$$

Add 1 to both sides.

$$7p^{2}+8p+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 8 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-8±\sqrt{8^{2}-4\times 7}}{2\times 7}$$

Square 8.

$$p=\frac{-8±\sqrt{64-4\times 7}}{2\times 7}$$

Multiply -4 times 7.

$$p=\frac{-8±\sqrt{64-28}}{2\times 7}$$

Add 64 to -28.

$$p=\frac{-8±\sqrt{36}}{2\times 7}$$

Take the square root of 36.

$$p=\frac{-8±6}{2\times 7}$$

Multiply 2 times 7.

$$p=\frac{-8±6}{14}$$

Now solve the equation p=\frac{-8±6}{14} when ± is plus. Add -8 to 6.

$$p=-\frac{2}{14}$$

Reduce the fraction \frac{-2}{14} to lowest terms by extracting and canceling out 2.

$$p=-\frac{1}{7}$$

Now solve the equation p=\frac{-8±6}{14} when ± is minus. Subtract 6 from -8.

$$p=-\frac{14}{14}$$

Divide -14 by 14.

$$p=-1$$

The equation is now solved.

$$p=-\frac{1}{7} p=-1$$

Steps for Completing the Square

Add 8p to both sides.

$$7p^{2}+8p=-1$$

Divide both sides by 7.

$$\frac{7p^{2}+8p}{7}=-\frac{1}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$p^{2}+\frac{8}{7}p=-\frac{1}{7}$$

Divide \frac{8}{7}, the coefficient of the x term, by 2 to get \frac{4}{7}. Then add the square of \frac{4}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}+\frac{8}{7}p+\left(\frac{4}{7}\right)^{2}=-\frac{1}{7}+\left(\frac{4}{7}\right)^{2}$$

Square \frac{4}{7} by squaring both the numerator and the denominator of the fraction.

$$p^{2}+\frac{8}{7}p+\frac{16}{49}=-\frac{1}{7}+\frac{16}{49}$$

Add -\frac{1}{7} to \frac{16}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p^{2}+\frac{8}{7}p+\frac{16}{49}=\frac{9}{49}$$

Factor p^{2}+\frac{8}{7}p+\frac{16}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p+\frac{4}{7}\right)^{2}=\frac{9}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p+\frac{4}{7}\right)^{2}}=\sqrt{\frac{9}{49}}$$

Simplify.

$$p+\frac{4}{7}=\frac{3}{7} p+\frac{4}{7}=-\frac{3}{7}$$

Subtract \frac{4}{7} from both sides of the equation.

$$p=-\frac{1}{7} p=-1$$