Solve p^2-22p=21

$$p ^ { 2 } - 22 p = 21$$

Solve for p

$$p=\sqrt{142}+11\approx 22.916375288$$

$$p=11-\sqrt{142}\approx -0.916375288$$

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.

$$p^{2}-22p=21$$

Subtract 21 from both sides of the equation.

$$p^{2}-22p-21=21-21$$

Subtracting 21 from itself leaves 0.

$$p^{2}-22p-21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -22 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\left(-21\right)}}{2}$$

Square -22.

$$p=\frac{-\left(-22\right)±\sqrt{484-4\left(-21\right)}}{2}$$

Multiply -4 times -21.

$$p=\frac{-\left(-22\right)±\sqrt{484+84}}{2}$$

Add 484 to 84.

$$p=\frac{-\left(-22\right)±\sqrt{568}}{2}$$

Take the square root of 568.

$$p=\frac{-\left(-22\right)±2\sqrt{142}}{2}$$

The opposite of -22 is 22.

$$p=\frac{22±2\sqrt{142}}{2}$$

Now solve the equation p=\frac{22±2\sqrt{142}}{2} when ± is plus. Add 22 to 2\sqrt{142}.

$$p=\frac{2\sqrt{142}+22}{2}$$

Divide 22+2\sqrt{142} by 2.

$$p=\sqrt{142}+11$$

Now solve the equation p=\frac{22±2\sqrt{142}}{2} when ± is minus. Subtract 2\sqrt{142} from 22.

$$p=\frac{22-2\sqrt{142}}{2}$$

Divide 22-2\sqrt{142} by 2.

$$p=11-\sqrt{142}$$

The equation is now solved.

$$p=\sqrt{142}+11 p=11-\sqrt{142}$$

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.

$$p^{2}-22p=21$$

Divide -22, the coefficient of the x term, by 2 to get -11. Then add the square of -11 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-22p+\left(-11\right)^{2}=21+\left(-11\right)^{2}$$

Square -11.

$$p^{2}-22p+121=21+121$$

Add 21 to 121.

$$p^{2}-22p+121=142$$

Factor p^{2}-22p+121. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p-11\right)^{2}=142$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-11\right)^{2}}=\sqrt{142}$$

Simplify.

$$p-11=\sqrt{142} p-11=-\sqrt{142}$$

Add 11 to both sides of the equation.

$$p=\sqrt{142}+11 p=11-\sqrt{142}$$