Solve 20p^2-15p=0

$$20 { p }^{ 2 } -15p = 0$$

Solve for p

$$p=\frac{3}{4}=0.75$$

$$p=0$$

Steps Using Factoring

Factor out p.

$$p\left(20p-15\right)=0$$

To find equation solutions, solve p=0 and 20p-15=0.

$$p=0 p=\frac{3}{4}$$

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.

$$20p^{2}-15p=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 20 for a, -15 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}}}{2\times 20}$$

Take the square root of \left(-15\right)^{2}.

$$p=\frac{-\left(-15\right)±15}{2\times 20}$$

The opposite of -15 is 15.

$$p=\frac{15±15}{2\times 20}$$

Multiply 2 times 20.

$$p=\frac{15±15}{40}$$

Now solve the equation p=\frac{15±15}{40} when ± is plus. Add 15 to 15.

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

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

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

Now solve the equation p=\frac{15±15}{40} when ± is minus. Subtract 15 from 15.

$$p=\frac{0}{40}$$

Divide 0 by 40.

$$p=0$$

The equation is now solved.

$$p=\frac{3}{4} p=0$$

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.

$$20p^{2}-15p=0$$

Divide both sides by 20.

$$\frac{20p^{2}-15p}{20}=\frac{0}{20}$$

Dividing by 20 undoes the multiplication by 20.

$$p^{2}+\left(-\frac{15}{20}\right)p=\frac{0}{20}$$

Reduce the fraction \frac{-15}{20} to lowest terms by extracting and canceling out 5.

$$p^{2}-\frac{3}{4}p=\frac{0}{20}$$

Divide 0 by 20.

$$p^{2}-\frac{3}{4}p=0$$

Divide -\frac{3}{4}, the coefficient of the x term, by 2 to get -\frac{3}{8}. Then add the square of -\frac{3}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-\frac{3}{4}p+\left(-\frac{3}{8}\right)^{2}=\left(-\frac{3}{8}\right)^{2}$$

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

$$p^{2}-\frac{3}{4}p+\frac{9}{64}=\frac{9}{64}$$

Factor p^{2}-\frac{3}{4}p+\frac{9}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p-\frac{3}{8}\right)^{2}=\frac{9}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-\frac{3}{8}\right)^{2}}=\sqrt{\frac{9}{64}}$$

Simplify.

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

Add \frac{3}{8} to both sides of the equation.

$$p=\frac{3}{4} p=0$$