Solve 88*20/300=(88+h)(30-h)/360

$$\frac { 88 \times 20 } { 300 } = \frac { ( 88 + h ) ( 30 - h ) } { 360 }$$

Solve for h

$$h=-66$$

$$h=8$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 1800, the least common multiple of 300,360.

$$6\times 88\times 20=5\left(88+h\right)\left(30-h\right)$$

Multiply 6 and 88 to get 528.

$$528\times 20=5\left(88+h\right)\left(30-h\right)$$

Multiply 528 and 20 to get 10560.

$$10560=5\left(88+h\right)\left(30-h\right)$$

Use the distributive property to multiply 5 by 88+h.

$$10560=\left(440+5h\right)\left(30-h\right)$$

Use the distributive property to multiply 440+5h by 30-h and combine like terms.

$$10560=13200-290h-5h^{2}$$

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

$$13200-290h-5h^{2}=10560$$

Subtract 10560 from both sides.

$$13200-290h-5h^{2}-10560=0$$

Subtract 10560 from 13200 to get 2640.

$$2640-290h-5h^{2}=0$$

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.

$$-5h^{2}-290h+2640=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -290 for b, and 2640 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$h=\frac{-\left(-290\right)±\sqrt{\left(-290\right)^{2}-4\left(-5\right)\times 2640}}{2\left(-5\right)}$$

Square -290.

$$h=\frac{-\left(-290\right)±\sqrt{84100-4\left(-5\right)\times 2640}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$h=\frac{-\left(-290\right)±\sqrt{84100+20\times 2640}}{2\left(-5\right)}$$

Multiply 20 times 2640.

$$h=\frac{-\left(-290\right)±\sqrt{84100+52800}}{2\left(-5\right)}$$

Add 84100 to 52800.

$$h=\frac{-\left(-290\right)±\sqrt{136900}}{2\left(-5\right)}$$

Take the square root of 136900.

$$h=\frac{-\left(-290\right)±370}{2\left(-5\right)}$$

The opposite of -290 is 290.

$$h=\frac{290±370}{2\left(-5\right)}$$

Multiply 2 times -5.

$$h=\frac{290±370}{-10}$$

Now solve the equation h=\frac{290±370}{-10} when ± is plus. Add 290 to 370.

$$h=\frac{660}{-10}$$

Divide 660 by -10.

$$h=-66$$

Now solve the equation h=\frac{290±370}{-10} when ± is minus. Subtract 370 from 290.

$$h=-\frac{80}{-10}$$

Divide -80 by -10.

$$h=8$$

The equation is now solved.

$$h=-66 h=8$$

Steps for Completing the Square

Multiply both sides of the equation by 1800, the least common multiple of 300,360.

$$6\times 88\times 20=5\left(88+h\right)\left(30-h\right)$$

Multiply 6 and 88 to get 528.

$$528\times 20=5\left(88+h\right)\left(30-h\right)$$

Multiply 528 and 20 to get 10560.

$$10560=5\left(88+h\right)\left(30-h\right)$$

Use the distributive property to multiply 5 by 88+h.

$$10560=\left(440+5h\right)\left(30-h\right)$$

Use the distributive property to multiply 440+5h by 30-h and combine like terms.

$$10560=13200-290h-5h^{2}$$

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

$$13200-290h-5h^{2}=10560$$

Subtract 13200 from both sides.

$$-290h-5h^{2}=10560-13200$$

Subtract 13200 from 10560 to get -2640.

$$-290h-5h^{2}=-2640$$

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.

$$-5h^{2}-290h=-2640$$

Divide both sides by -5.

$$\frac{-5h^{2}-290h}{-5}=-\frac{2640}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$h^{2}+\left(-\frac{290}{-5}\right)h=-\frac{2640}{-5}$$

Divide -290 by -5.

$$h^{2}+58h=-\frac{2640}{-5}$$

Divide -2640 by -5.

$$h^{2}+58h=528$$

Divide 58, the coefficient of the x term, by 2 to get 29. Then add the square of 29 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$h^{2}+58h+29^{2}=528+29^{2}$$

Square 29.

$$h^{2}+58h+841=528+841$$

Add 528 to 841.

$$h^{2}+58h+841=1369$$

Factor h^{2}+58h+841. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(h+29\right)^{2}=1369$$

Take the square root of both sides of the equation.

$$\sqrt{\left(h+29\right)^{2}}=\sqrt{1369}$$

Simplify.

$$h+29=37 h+29=-37$$

Subtract 29 from both sides of the equation.

$$h=8 h=-66$$