Solve 72=h^2+23h

$$72 = h ^ { 2 } + 23 h$$

Solve for h

$$h = \frac{\sqrt{817} - 23}{2} \approx 2.791605928$$

$$h=\frac{-\sqrt{817}-23}{2}\approx -25.791605928$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$h^{2}+23h=72$$

Subtract 72 from both sides.

$$h^{2}+23h-72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 23 for b, and -72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$h=\frac{-23±\sqrt{23^{2}-4\left(-72\right)}}{2}$$

Square 23.

$$h=\frac{-23±\sqrt{529-4\left(-72\right)}}{2}$$

Multiply -4 times -72.

$$h=\frac{-23±\sqrt{529+288}}{2}$$

Add 529 to 288.

$$h=\frac{-23±\sqrt{817}}{2}$$

Now solve the equation h=\frac{-23±\sqrt{817}}{2} when ± is plus. Add -23 to \sqrt{817}.

$$h=\frac{\sqrt{817}-23}{2}$$

Now solve the equation h=\frac{-23±\sqrt{817}}{2} when ± is minus. Subtract \sqrt{817} from -23.

$$h=\frac{-\sqrt{817}-23}{2}$$

The equation is now solved.

$$h=\frac{\sqrt{817}-23}{2} h=\frac{-\sqrt{817}-23}{2}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$h^{2}+23h=72$$

Divide 23, the coefficient of the x term, by 2 to get \frac{23}{2}. Then add the square of \frac{23}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$h^{2}+23h+\left(\frac{23}{2}\right)^{2}=72+\left(\frac{23}{2}\right)^{2}$$

Square \frac{23}{2} by squaring both the numerator and the denominator of the fraction.

$$h^{2}+23h+\frac{529}{4}=72+\frac{529}{4}$$

Add 72 to \frac{529}{4}.

$$h^{2}+23h+\frac{529}{4}=\frac{817}{4}$$

Factor h^{2}+23h+\frac{529}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(h+\frac{23}{2}\right)^{2}=\frac{817}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(h+\frac{23}{2}\right)^{2}}=\sqrt{\frac{817}{4}}$$

Simplify.

$$h+\frac{23}{2}=\frac{\sqrt{817}}{2} h+\frac{23}{2}=-\frac{\sqrt{817}}{2}$$

Subtract \frac{23}{2} from both sides of the equation.

$$h=\frac{\sqrt{817}-23}{2} h=\frac{-\sqrt{817}-23}{2}$$