Solve 50/17*9800+34*9800h=26500(h^2-8.875^2)

$$\frac { 50 } { 17 } \times 9800 + 34 \times 9800 h = 26500 ( h ^ { 2 } - 8.875 ^ { 2 } )$$

Solve for h

$$h=\frac{\sqrt{155056672801}}{36040}+\frac{1666}{265}\approx 17.212773631$$

$$h=-\frac{\sqrt{155056672801}}{36040}+\frac{1666}{265}\approx -4.639188725$$

Steps Using the Quadratic Formula

Multiply \frac{50}{17} and 9800 to get \frac{490000}{17}.

$$\frac{490000}{17}+34\times 9800h=26500\left(h^{2}-8.875^{2}\right)$$

Multiply 34 and 9800 to get 333200.

$$\frac{490000}{17}+333200h=26500\left(h^{2}-8.875^{2}\right)$$

Calculate 8.875 to the power of 2 and get 78.765625.

$$\frac{490000}{17}+333200h=26500\left(h^{2}-78.765625\right)$$

Use the distributive property to multiply 26500 by h^{2}-78.765625.

$$\frac{490000}{17}+333200h=26500h^{2}-2087289.0625$$

Subtract 26500h^{2} from both sides.

$$\frac{490000}{17}+333200h-26500h^{2}=-2087289.0625$$

Add 2087289.0625 to both sides.

$$\frac{490000}{17}+333200h-26500h^{2}+2087289.0625=0$$

Add \frac{490000}{17} and 2087289.0625 to get \frac{575582625}{272}.

$$\frac{575582625}{272}+333200h-26500h^{2}=0$$

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.

$$-26500h^{2}+333200h+\frac{575582625}{272}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -26500 for a, 333200 for b, and \frac{575582625}{272} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$h=\frac{-333200±\sqrt{333200^{2}-4\left(-26500\right)\times \frac{575582625}{272}}}{2\left(-26500\right)}$$

Square 333200.

$$h=\frac{-333200±\sqrt{111022240000-4\left(-26500\right)\times \frac{575582625}{272}}}{2\left(-26500\right)}$$

Multiply -4 times -26500.

$$h=\frac{-333200±\sqrt{111022240000+106000\times \frac{575582625}{272}}}{2\left(-26500\right)}$$

Multiply 106000 times \frac{575582625}{272}.

$$h=\frac{-333200±\sqrt{111022240000+\frac{3813234890625}{17}}}{2\left(-26500\right)}$$

Add 111022240000 to \frac{3813234890625}{17}.

$$h=\frac{-333200±\sqrt{\frac{5700612970625}{17}}}{2\left(-26500\right)}$$

Take the square root of \frac{5700612970625}{17}.

$$h=\frac{-333200±\frac{25\sqrt{155056672801}}{17}}{2\left(-26500\right)}$$

Multiply 2 times -26500.

$$h=\frac{-333200±\frac{25\sqrt{155056672801}}{17}}{-53000}$$

Now solve the equation h=\frac{-333200±\frac{25\sqrt{155056672801}}{17}}{-53000} when ± is plus. Add -333200 to \frac{25\sqrt{155056672801}}{17}.

$$h=\frac{\frac{25\sqrt{155056672801}}{17}-333200}{-53000}$$

Divide -333200+\frac{25\sqrt{155056672801}}{17} by -53000.

$$h=-\frac{\sqrt{155056672801}}{36040}+\frac{1666}{265}$$

Now solve the equation h=\frac{-333200±\frac{25\sqrt{155056672801}}{17}}{-53000} when ± is minus. Subtract \frac{25\sqrt{155056672801}}{17} from -333200.

$$h=\frac{-\frac{25\sqrt{155056672801}}{17}-333200}{-53000}$$

Divide -333200-\frac{25\sqrt{155056672801}}{17} by -53000.

$$h=\frac{\sqrt{155056672801}}{36040}+\frac{1666}{265}$$

The equation is now solved.

$$h=-\frac{\sqrt{155056672801}}{36040}+\frac{1666}{265} h=\frac{\sqrt{155056672801}}{36040}+\frac{1666}{265}$$

Steps for Completing the Square

Multiply \frac{50}{17} and 9800 to get \frac{490000}{17}.

$$\frac{490000}{17}+34\times 9800h=26500\left(h^{2}-8.875^{2}\right)$$

Multiply 34 and 9800 to get 333200.

$$\frac{490000}{17}+333200h=26500\left(h^{2}-8.875^{2}\right)$$

Calculate 8.875 to the power of 2 and get 78.765625.

$$\frac{490000}{17}+333200h=26500\left(h^{2}-78.765625\right)$$

Use the distributive property to multiply 26500 by h^{2}-78.765625.

$$\frac{490000}{17}+333200h=26500h^{2}-2087289.0625$$

Subtract 26500h^{2} from both sides.

$$\frac{490000}{17}+333200h-26500h^{2}=-2087289.0625$$

Subtract \frac{490000}{17} from both sides.

$$333200h-26500h^{2}=-2087289.0625-\frac{490000}{17}$$

Subtract \frac{490000}{17} from -2087289.0625 to get -\frac{575582625}{272}.

$$333200h-26500h^{2}=-\frac{575582625}{272}$$

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.

$$-26500h^{2}+333200h=-\frac{575582625}{272}$$

Divide both sides by -26500.

$$\frac{-26500h^{2}+333200h}{-26500}=-\frac{\frac{575582625}{272}}{-26500}$$

Dividing by -26500 undoes the multiplication by -26500.

$$h^{2}+\frac{333200}{-26500}h=-\frac{\frac{575582625}{272}}{-26500}$$

Reduce the fraction \frac{333200}{-26500} to lowest terms by extracting and canceling out 100.

$$h^{2}-\frac{3332}{265}h=-\frac{\frac{575582625}{272}}{-26500}$$

Divide -\frac{575582625}{272} by -26500.

$$h^{2}-\frac{3332}{265}h=\frac{4604661}{57664}$$

Divide -\frac{3332}{265}, the coefficient of the x term, by 2 to get -\frac{1666}{265}. Then add the square of -\frac{1666}{265} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$h^{2}-\frac{3332}{265}h+\left(-\frac{1666}{265}\right)^{2}=\frac{4604661}{57664}+\left(-\frac{1666}{265}\right)^{2}$$

Square -\frac{1666}{265} by squaring both the numerator and the denominator of the fraction.

$$h^{2}-\frac{3332}{265}h+\frac{2775556}{70225}=\frac{4604661}{57664}+\frac{2775556}{70225}$$

Add \frac{4604661}{57664} to \frac{2775556}{70225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$h^{2}-\frac{3332}{265}h+\frac{2775556}{70225}=\frac{9120980753}{76404800}$$

Factor h^{2}-\frac{3332}{265}h+\frac{2775556}{70225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(h-\frac{1666}{265}\right)^{2}=\frac{9120980753}{76404800}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(h-\frac{1666}{265}\right)^{2}}=\sqrt{\frac{9120980753}{76404800}}$$

Simplify.

$$h-\frac{1666}{265}=\frac{\sqrt{155056672801}}{36040} h-\frac{1666}{265}=-\frac{\sqrt{155056672801}}{36040}$$

Add \frac{1666}{265} to both sides of the equation.

$$h=\frac{\sqrt{155056672801}}{36040}+\frac{1666}{265} h=-\frac{\sqrt{155056672801}}{36040}+\frac{1666}{265}$$