Solve 15p^2+80=-80p

$$15 p ^ { 2 } + 80 = - 80 p$$

Solve for p

$$p = -\frac{4}{3} = -1\frac{1}{3} \approx -1.333333333$$

$$p=-4$$

Steps Using the Quadratic Formula

Add 80p to both sides.

$$15p^{2}+80+80p=0$$

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.

$$15p^{2}+80p+80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, 80 for b, and 80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-80±\sqrt{80^{2}-4\times 15\times 80}}{2\times 15}$$

Square 80.

$$p=\frac{-80±\sqrt{6400-4\times 15\times 80}}{2\times 15}$$

Multiply -4 times 15.

$$p=\frac{-80±\sqrt{6400-60\times 80}}{2\times 15}$$

Multiply -60 times 80.

$$p=\frac{-80±\sqrt{6400-4800}}{2\times 15}$$

Add 6400 to -4800.

$$p=\frac{-80±\sqrt{1600}}{2\times 15}$$

Take the square root of 1600.

$$p=\frac{-80±40}{2\times 15}$$

Multiply 2 times 15.

$$p=\frac{-80±40}{30}$$

Now solve the equation p=\frac{-80±40}{30} when ± is plus. Add -80 to 40.

$$p=-\frac{40}{30}$$

Reduce the fraction \frac{-40}{30} to lowest terms by extracting and canceling out 10.

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

Now solve the equation p=\frac{-80±40}{30} when ± is minus. Subtract 40 from -80.

$$p=-\frac{120}{30}$$

Divide -120 by 30.

$$p=-4$$

The equation is now solved.

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

Steps for Completing the Square

Add 80p to both sides.

$$15p^{2}+80+80p=0$$

Subtract 80 from both sides. Anything subtracted from zero gives its negation.

$$15p^{2}+80p=-80$$

Divide both sides by 15.

$$\frac{15p^{2}+80p}{15}=-\frac{80}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$p^{2}+\frac{80}{15}p=-\frac{80}{15}$$

Reduce the fraction \frac{80}{15} to lowest terms by extracting and canceling out 5.

$$p^{2}+\frac{16}{3}p=-\frac{80}{15}$$

Reduce the fraction \frac{-80}{15} to lowest terms by extracting and canceling out 5.

$$p^{2}+\frac{16}{3}p=-\frac{16}{3}$$

Divide \frac{16}{3}, the coefficient of the x term, by 2 to get \frac{8}{3}. Then add the square of \frac{8}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}+\frac{16}{3}p+\left(\frac{8}{3}\right)^{2}=-\frac{16}{3}+\left(\frac{8}{3}\right)^{2}$$

Square \frac{8}{3} by squaring both the numerator and the denominator of the fraction.

$$p^{2}+\frac{16}{3}p+\frac{64}{9}=-\frac{16}{3}+\frac{64}{9}$$

Add -\frac{16}{3} to \frac{64}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p^{2}+\frac{16}{3}p+\frac{64}{9}=\frac{16}{9}$$

Factor p^{2}+\frac{16}{3}p+\frac{64}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p+\frac{8}{3}\right)^{2}=\frac{16}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p+\frac{8}{3}\right)^{2}}=\sqrt{\frac{16}{9}}$$

Simplify.

$$p+\frac{8}{3}=\frac{4}{3} p+\frac{8}{3}=-\frac{4}{3}$$

Subtract \frac{8}{3} from both sides of the equation.

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