Solve 31p^2-37p=0

$$31 p ^ { 2 } - 37 p = 0$$

Solve for p

$$p = \frac{37}{31} = 1\frac{6}{31} \approx 1.193548387$$

$$p=0$$

Steps Using Factoring

Factor out p.

$$p\left(31p-37\right)=0$$

To find equation solutions, solve p=0 and 31p-37=0.

$$p=0 p=\frac{37}{31}$$

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.

$$31p^{2}-37p=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 31 for a, -37 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-37\right)±\sqrt{\left(-37\right)^{2}}}{2\times 31}$$

Take the square root of \left(-37\right)^{2}.

$$p=\frac{-\left(-37\right)±37}{2\times 31}$$

The opposite of -37 is 37.

$$p=\frac{37±37}{2\times 31}$$

Multiply 2 times 31.

$$p=\frac{37±37}{62}$$

Now solve the equation p=\frac{37±37}{62} when ± is plus. Add 37 to 37.

$$p=\frac{74}{62}$$

Reduce the fraction \frac{74}{62} to lowest terms by extracting and canceling out 2.

$$p=\frac{37}{31}$$

Now solve the equation p=\frac{37±37}{62} when ± is minus. Subtract 37 from 37.

$$p=\frac{0}{62}$$

Divide 0 by 62.

$$p=0$$

The equation is now solved.

$$p=\frac{37}{31} p=0$$

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.

$$31p^{2}-37p=0$$

Divide both sides by 31.

$$\frac{31p^{2}-37p}{31}=\frac{0}{31}$$

Dividing by 31 undoes the multiplication by 31.

$$p^{2}-\frac{37}{31}p=\frac{0}{31}$$

Divide 0 by 31.

$$p^{2}-\frac{37}{31}p=0$$

Divide -\frac{37}{31}, the coefficient of the x term, by 2 to get -\frac{37}{62}. Then add the square of -\frac{37}{62} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-\frac{37}{31}p+\left(-\frac{37}{62}\right)^{2}=\left(-\frac{37}{62}\right)^{2}$$

Square -\frac{37}{62} by squaring both the numerator and the denominator of the fraction.

$$p^{2}-\frac{37}{31}p+\frac{1369}{3844}=\frac{1369}{3844}$$

Factor p^{2}-\frac{37}{31}p+\frac{1369}{3844}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p-\frac{37}{62}\right)^{2}=\frac{1369}{3844}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-\frac{37}{62}\right)^{2}}=\sqrt{\frac{1369}{3844}}$$

Simplify.

$$p-\frac{37}{62}=\frac{37}{62} p-\frac{37}{62}=-\frac{37}{62}$$

Add \frac{37}{62} to both sides of the equation.

$$p=\frac{37}{31} p=0$$