Solve -h^2+3h+7=4h-7

$$- h ^ { 2 } + 3 h + 7 = 4 h - 7$$

Solve for h

$$h = \frac{\sqrt{57} - 1}{2} \approx 3.274917218$$

$$h=\frac{-\sqrt{57}-1}{2}\approx -4.274917218$$

Steps Using the Quadratic Formula

Subtract 4h from both sides.

$$-h^{2}+3h+7-4h=-7$$

Combine 3h and -4h to get -h.

$$-h^{2}-h+7=-7$$

Add 7 to both sides.

$$-h^{2}-h+7+7=0$$

Add 7 and 7 to get 14.

$$-h^{2}-h+14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -1 for b, and 14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$h=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)\times 14}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$h=\frac{-\left(-1\right)±\sqrt{1+4\times 14}}{2\left(-1\right)}$$

Multiply 4 times 14.

$$h=\frac{-\left(-1\right)±\sqrt{1+56}}{2\left(-1\right)}$$

Add 1 to 56.

$$h=\frac{-\left(-1\right)±\sqrt{57}}{2\left(-1\right)}$$

The opposite of -1 is 1.

$$h=\frac{1±\sqrt{57}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$h=\frac{1±\sqrt{57}}{-2}$$

Now solve the equation h=\frac{1±\sqrt{57}}{-2} when ± is plus. Add 1 to \sqrt{57}.

$$h=\frac{\sqrt{57}+1}{-2}$$

Divide 1+\sqrt{57} by -2.

$$h=\frac{-\sqrt{57}-1}{2}$$

Now solve the equation h=\frac{1±\sqrt{57}}{-2} when ± is minus. Subtract \sqrt{57} from 1.

$$h=\frac{1-\sqrt{57}}{-2}$$

Divide 1-\sqrt{57} by -2.

$$h=\frac{\sqrt{57}-1}{2}$$

The equation is now solved.

$$h=\frac{-\sqrt{57}-1}{2} h=\frac{\sqrt{57}-1}{2}$$

Steps for Completing the Square

Subtract 4h from both sides.

$$-h^{2}+3h+7-4h=-7$$

Combine 3h and -4h to get -h.

$$-h^{2}-h+7=-7$$

Subtract 7 from both sides.

$$-h^{2}-h=-7-7$$

Subtract 7 from -7 to get -14.

$$-h^{2}-h=-14$$

Divide both sides by -1.

$$\frac{-h^{2}-h}{-1}=-\frac{14}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$h^{2}+\left(-\frac{1}{-1}\right)h=-\frac{14}{-1}$$

Divide -1 by -1.

$$h^{2}+h=-\frac{14}{-1}$$

Divide -14 by -1.

$$h^{2}+h=14$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$h^{2}+h+\left(\frac{1}{2}\right)^{2}=14+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$h^{2}+h+\frac{1}{4}=14+\frac{1}{4}$$

Add 14 to \frac{1}{4}.

$$h^{2}+h+\frac{1}{4}=\frac{57}{4}$$

Factor h^{2}+h+\frac{1}{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}.

$$\left(h+\frac{1}{2}\right)^{2}=\frac{57}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(h+\frac{1}{2}\right)^{2}}=\sqrt{\frac{57}{4}}$$

Simplify.

$$h+\frac{1}{2}=\frac{\sqrt{57}}{2} h+\frac{1}{2}=-\frac{\sqrt{57}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$h=\frac{\sqrt{57}-1}{2} h=\frac{-\sqrt{57}-1}{2}$$