Solve 1600h^2-360h=9/12^-111

$$1600 { h }^{ 2 } -360h = \frac{ 9 }{ { 12 }^{ -1 } } 11$$

Solve for h

$$h=\frac{3\sqrt{537}+9}{80}\approx 0.981497267$$

$$h=\frac{9-3\sqrt{537}}{80}\approx -0.756497267$$

Steps Using the Quadratic Formula

Calculate 12 to the power of -1 and get \frac{1}{12}.

$$1600h^{2}-360h=\frac{9}{\frac{1}{12}}\times 11$$

Divide 9 by \frac{1}{12} by multiplying 9 by the reciprocal of \frac{1}{12}.

$$1600h^{2}-360h=9\times 12\times 11$$

Multiply 9 and 12 to get 108.

$$1600h^{2}-360h=108\times 11$$

Multiply 108 and 11 to get 1188.

$$1600h^{2}-360h=1188$$

Subtract 1188 from both sides.

$$1600h^{2}-360h-1188=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1600 for a, -360 for b, and -1188 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$h=\frac{-\left(-360\right)±\sqrt{\left(-360\right)^{2}-4\times 1600\left(-1188\right)}}{2\times 1600}$$

Square -360.

$$h=\frac{-\left(-360\right)±\sqrt{129600-4\times 1600\left(-1188\right)}}{2\times 1600}$$

Multiply -4 times 1600.

$$h=\frac{-\left(-360\right)±\sqrt{129600-6400\left(-1188\right)}}{2\times 1600}$$

Multiply -6400 times -1188.

$$h=\frac{-\left(-360\right)±\sqrt{129600+7603200}}{2\times 1600}$$

Add 129600 to 7603200.

$$h=\frac{-\left(-360\right)±\sqrt{7732800}}{2\times 1600}$$

Take the square root of 7732800.

$$h=\frac{-\left(-360\right)±120\sqrt{537}}{2\times 1600}$$

The opposite of -360 is 360.

$$h=\frac{360±120\sqrt{537}}{2\times 1600}$$

Multiply 2 times 1600.

$$h=\frac{360±120\sqrt{537}}{3200}$$

Now solve the equation h=\frac{360±120\sqrt{537}}{3200} when ± is plus. Add 360 to 120\sqrt{537}.

$$h=\frac{120\sqrt{537}+360}{3200}$$

Divide 360+120\sqrt{537} by 3200.

$$h=\frac{3\sqrt{537}+9}{80}$$

Now solve the equation h=\frac{360±120\sqrt{537}}{3200} when ± is minus. Subtract 120\sqrt{537} from 360.

$$h=\frac{360-120\sqrt{537}}{3200}$$

Divide 360-120\sqrt{537} by 3200.

$$h=\frac{9-3\sqrt{537}}{80}$$

The equation is now solved.

$$h=\frac{3\sqrt{537}+9}{80} h=\frac{9-3\sqrt{537}}{80}$$

Steps for Completing the Square

Calculate 12 to the power of -1 and get \frac{1}{12}.

$$1600h^{2}-360h=\frac{9}{\frac{1}{12}}\times 11$$

Divide 9 by \frac{1}{12} by multiplying 9 by the reciprocal of \frac{1}{12}.

$$1600h^{2}-360h=9\times 12\times 11$$

Multiply 9 and 12 to get 108.

$$1600h^{2}-360h=108\times 11$$

Multiply 108 and 11 to get 1188.

$$1600h^{2}-360h=1188$$

Divide both sides by 1600.

$$\frac{1600h^{2}-360h}{1600}=\frac{1188}{1600}$$

Dividing by 1600 undoes the multiplication by 1600.

$$h^{2}+\left(-\frac{360}{1600}\right)h=\frac{1188}{1600}$$

Reduce the fraction \frac{-360}{1600} to lowest terms by extracting and canceling out 40.

$$h^{2}-\frac{9}{40}h=\frac{1188}{1600}$$

Reduce the fraction \frac{1188}{1600} to lowest terms by extracting and canceling out 4.

$$h^{2}-\frac{9}{40}h=\frac{297}{400}$$

Divide -\frac{9}{40}, the coefficient of the x term, by 2 to get -\frac{9}{80}. Then add the square of -\frac{9}{80} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$h^{2}-\frac{9}{40}h+\left(-\frac{9}{80}\right)^{2}=\frac{297}{400}+\left(-\frac{9}{80}\right)^{2}$$

Square -\frac{9}{80} by squaring both the numerator and the denominator of the fraction.

$$h^{2}-\frac{9}{40}h+\frac{81}{6400}=\frac{297}{400}+\frac{81}{6400}$$

Add \frac{297}{400} to \frac{81}{6400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$h^{2}-\frac{9}{40}h+\frac{81}{6400}=\frac{4833}{6400}$$

Factor h^{2}-\frac{9}{40}h+\frac{81}{6400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(h-\frac{9}{80}\right)^{2}=\frac{4833}{6400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(h-\frac{9}{80}\right)^{2}}=\sqrt{\frac{4833}{6400}}$$

Simplify.

$$h-\frac{9}{80}=\frac{3\sqrt{537}}{80} h-\frac{9}{80}=-\frac{3\sqrt{537}}{80}$$

Add \frac{9}{80} to both sides of the equation.

$$h=\frac{3\sqrt{537}+9}{80} h=\frac{9-3\sqrt{537}}{80}$$