Solve 20left(left(x-2right)left(60-x-2right)-16right)=14240

$$20 \left( \left( x-2 \right) \left( 60-x-2 \right) -16 \right) = 14240$$

Solve for x

$$x=2\sqrt{14}+30\approx 37.483314774$$

$$x=30-2\sqrt{14}\approx 22.516685226$$

Steps Using the Quadratic Formula

Divide both sides by 20.

$$\left(x-2\right)\left(60-x-2\right)-16=\frac{14240}{20}$$

Divide 14240 by 20 to get 712.

$$\left(x-2\right)\left(60-x-2\right)-16=712$$

Subtract 2 from 60 to get 58.

$$\left(x-2\right)\left(58-x\right)-16=712$$

Use the distributive property to multiply x-2 by 58-x and combine like terms.

$$60x-x^{2}-116-16=712$$

Subtract 16 from -116 to get -132.

$$60x-x^{2}-132=712$$

Subtract 712 from both sides.

$$60x-x^{2}-132-712=0$$

Subtract 712 from -132 to get -844.

$$60x-x^{2}-844=0$$

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.

$$-x^{2}+60x-844=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 60 for b, and -844 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\left(-1\right)\left(-844\right)}}{2\left(-1\right)}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\left(-1\right)\left(-844\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-60±\sqrt{3600+4\left(-844\right)}}{2\left(-1\right)}$$

Multiply 4 times -844.

$$x=\frac{-60±\sqrt{3600-3376}}{2\left(-1\right)}$$

Add 3600 to -3376.

$$x=\frac{-60±\sqrt{224}}{2\left(-1\right)}$$

Take the square root of 224.

$$x=\frac{-60±4\sqrt{14}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-60±4\sqrt{14}}{-2}$$

Now solve the equation x=\frac{-60±4\sqrt{14}}{-2} when ± is plus. Add -60 to 4\sqrt{14}.

$$x=\frac{4\sqrt{14}-60}{-2}$$

Divide -60+4\sqrt{14} by -2.

$$x=30-2\sqrt{14}$$

Now solve the equation x=\frac{-60±4\sqrt{14}}{-2} when ± is minus. Subtract 4\sqrt{14} from -60.

$$x=\frac{-4\sqrt{14}-60}{-2}$$

Divide -60-4\sqrt{14} by -2.

$$x=2\sqrt{14}+30$$

The equation is now solved.

$$x=30-2\sqrt{14} x=2\sqrt{14}+30$$

Steps for Completing the Square

Divide both sides by 20.

$$\left(x-2\right)\left(60-x-2\right)-16=\frac{14240}{20}$$

Divide 14240 by 20 to get 712.

$$\left(x-2\right)\left(60-x-2\right)-16=712$$

Subtract 2 from 60 to get 58.

$$\left(x-2\right)\left(58-x\right)-16=712$$

Use the distributive property to multiply x-2 by 58-x and combine like terms.

$$60x-x^{2}-116-16=712$$

Subtract 16 from -116 to get -132.

$$60x-x^{2}-132=712$$

Add 132 to both sides.

$$60x-x^{2}=712+132$$

Add 712 and 132 to get 844.

$$60x-x^{2}=844$$

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.

$$-x^{2}+60x=844$$

Divide both sides by -1.

$$\frac{-x^{2}+60x}{-1}=\frac{844}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{60}{-1}x=\frac{844}{-1}$$

Divide 60 by -1.

$$x^{2}-60x=\frac{844}{-1}$$

Divide 844 by -1.

$$x^{2}-60x=-844$$

Divide -60, the coefficient of the x term, by 2 to get -30. Then add the square of -30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-60x+\left(-30\right)^{2}=-844+\left(-30\right)^{2}$$

Square -30.

$$x^{2}-60x+900=-844+900$$

Add -844 to 900.

$$x^{2}-60x+900=56$$

Factor x^{2}-60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-30\right)^{2}=56$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-30\right)^{2}}=\sqrt{56}$$

Simplify.

$$x-30=2\sqrt{14} x-30=-2\sqrt{14}$$

Add 30 to both sides of the equation.

$$x=2\sqrt{14}+30 x=30-2\sqrt{14}$$