Solve 16p^2+8p-8=0

$$16 { p }^{ 2 } +8p-8 = 0$$

Solve for p

$$p=-1$$

$$p=\frac{1}{2}=0.5$$

Steps Using Factoring By Grouping

Divide both sides by 8.

$$2p^{2}+p-1=0$$

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

$$a+b=1 ab=2\left(-1\right)=-2$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=-1 b=2$$

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

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

Factor out p in 2p^{2}-p.

$$p\left(2p-1\right)+2p-1$$

Factor out common term 2p-1 by using distributive property.

$$\left(2p-1\right)\left(p+1\right)$$

To find equation solutions, solve 2p-1=0 and p+1=0.

$$p=\frac{1}{2} p=-1$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$16p^{2}+8p-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 8 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-8±\sqrt{8^{2}-4\times 16\left(-8\right)}}{2\times 16}$$

Square 8.

$$p=\frac{-8±\sqrt{64-4\times 16\left(-8\right)}}{2\times 16}$$

Multiply -4 times 16.

$$p=\frac{-8±\sqrt{64-64\left(-8\right)}}{2\times 16}$$

Multiply -64 times -8.

$$p=\frac{-8±\sqrt{64+512}}{2\times 16}$$

Add 64 to 512.

$$p=\frac{-8±\sqrt{576}}{2\times 16}$$

Take the square root of 576.

$$p=\frac{-8±24}{2\times 16}$$

Multiply 2 times 16.

$$p=\frac{-8±24}{32}$$

Now solve the equation p=\frac{-8±24}{32} when ± is plus. Add -8 to 24.

$$p=\frac{16}{32}$$

Reduce the fraction \frac{16}{32} to lowest terms by extracting and canceling out 16.

$$p=\frac{1}{2}$$

Now solve the equation p=\frac{-8±24}{32} when ± is minus. Subtract 24 from -8.

$$p=-\frac{32}{32}$$

Divide -32 by 32.

$$p=-1$$

The equation is now solved.

$$p=\frac{1}{2} p=-1$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$16p^{2}+8p-8=0$$

Add 8 to both sides of the equation.

$$16p^{2}+8p-8-\left(-8\right)=-\left(-8\right)$$

Subtracting -8 from itself leaves 0.

$$16p^{2}+8p=-\left(-8\right)$$

Subtract -8 from 0.

$$16p^{2}+8p=8$$

Divide both sides by 16.

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

Dividing by 16 undoes the multiplication by 16.

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

Reduce the fraction \frac{8}{16} to lowest terms by extracting and canceling out 8.

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

Reduce the fraction \frac{8}{16} to lowest terms by extracting and canceling out 8.

$$p^{2}+\frac{1}{2}p=\frac{1}{2}$$

Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$p^{2}+\frac{1}{2}p+\frac{1}{16}=\frac{1}{2}+\frac{1}{16}$$

Add \frac{1}{2} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p^{2}+\frac{1}{2}p+\frac{1}{16}=\frac{9}{16}$$

Factor p^{2}+\frac{1}{2}p+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p+\frac{1}{4}\right)^{2}=\frac{9}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p+\frac{1}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Simplify.

$$p+\frac{1}{4}=\frac{3}{4} p+\frac{1}{4}=-\frac{3}{4}$$

Subtract \frac{1}{4} from both sides of the equation.

$$p=\frac{1}{2} p=-1$$