Solve (x-20)[600-10(x-25)]=5500

$$( x - 20 ) [ 600 - 10 ( x - 25 ) ] = 5500$$

Solve for x

$$x=30$$

$$x=75$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -10 by x-25.

$$\left(x-20\right)\left(600-10x+250\right)=5500$$

Add 600 and 250 to get 850.

$$\left(x-20\right)\left(850-10x\right)=5500$$

Apply the distributive property by multiplying each term of x-20 by each term of 850-10x.

$$850x-10x^{2}-17000+200x=5500$$

Combine 850x and 200x to get 1050x.

$$1050x-10x^{2}-17000=5500$$

Subtract 5500 from both sides.

$$1050x-10x^{2}-17000-5500=0$$

Subtract 5500 from -17000 to get -22500.

$$1050x-10x^{2}-22500=0$$

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.

$$-10x^{2}+1050x-22500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 1050 for b, and -22500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1050±\sqrt{1050^{2}-4\left(-10\right)\left(-22500\right)}}{2\left(-10\right)}$$

Square 1050.

$$x=\frac{-1050±\sqrt{1102500-4\left(-10\right)\left(-22500\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-1050±\sqrt{1102500+40\left(-22500\right)}}{2\left(-10\right)}$$

Multiply 40 times -22500.

$$x=\frac{-1050±\sqrt{1102500-900000}}{2\left(-10\right)}$$

Add 1102500 to -900000.

$$x=\frac{-1050±\sqrt{202500}}{2\left(-10\right)}$$

Take the square root of 202500.

$$x=\frac{-1050±450}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-1050±450}{-20}$$

Now solve the equation x=\frac{-1050±450}{-20} when ± is plus. Add -1050 to 450.

$$x=-\frac{600}{-20}$$

Divide -600 by -20.

$$x=30$$

Now solve the equation x=\frac{-1050±450}{-20} when ± is minus. Subtract 450 from -1050.

$$x=-\frac{1500}{-20}$$

Divide -1500 by -20.

$$x=75$$

The equation is now solved.

$$x=30 x=75$$

Steps for Completing the Square

Use the distributive property to multiply -10 by x-25.

$$\left(x-20\right)\left(600-10x+250\right)=5500$$

Add 600 and 250 to get 850.

$$\left(x-20\right)\left(850-10x\right)=5500$$

Apply the distributive property by multiplying each term of x-20 by each term of 850-10x.

$$850x-10x^{2}-17000+200x=5500$$

Combine 850x and 200x to get 1050x.

$$1050x-10x^{2}-17000=5500$$

Add 17000 to both sides.

$$1050x-10x^{2}=5500+17000$$

Add 5500 and 17000 to get 22500.

$$1050x-10x^{2}=22500$$

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.

$$-10x^{2}+1050x=22500$$

Divide both sides by -10.

$$\frac{-10x^{2}+1050x}{-10}=\frac{22500}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{1050}{-10}x=\frac{22500}{-10}$$

Divide 1050 by -10.

$$x^{2}-105x=\frac{22500}{-10}$$

Divide 22500 by -10.

$$x^{2}-105x=-2250$$

Divide -105, the coefficient of the x term, by 2 to get -\frac{105}{2}. Then add the square of -\frac{105}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-105x+\left(-\frac{105}{2}\right)^{2}=-2250+\left(-\frac{105}{2}\right)^{2}$$

Square -\frac{105}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-105x+\frac{11025}{4}=-2250+\frac{11025}{4}$$

Add -2250 to \frac{11025}{4}.

$$x^{2}-105x+\frac{11025}{4}=\frac{2025}{4}$$

Factor x^{2}-105x+\frac{11025}{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}.

$$\left(x-\frac{105}{2}\right)^{2}=\frac{2025}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{105}{2}\right)^{2}}=\sqrt{\frac{2025}{4}}$$

Simplify.

$$x-\frac{105}{2}=\frac{45}{2} x-\frac{105}{2}=-\frac{45}{2}$$

Add \frac{105}{2} to both sides of the equation.

$$x=75 x=30$$