Solve (x-20)[600-10(x-25)]=550

$$( x - 20 ) [ 600 - 10 ( x - 25 ) ] = 550$$

Solve for x

$$x = \frac{3 \sqrt{445} + 105}{2} \approx 84.142534665$$

$$x = \frac{105 - 3 \sqrt{445}}{2} \approx 20.857465335$$

Steps Using the Quadratic Formula

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

$$\left(x-20\right)\left(600-10x+250\right)=550$$

Add 600 and 250 to get 850.

$$\left(x-20\right)\left(850-10x\right)=550$$

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

$$850x-10x^{2}-17000+200x=550$$

Combine 850x and 200x to get 1050x.

$$1050x-10x^{2}-17000=550$$

Subtract 550 from both sides.

$$1050x-10x^{2}-17000-550=0$$

Subtract 550 from -17000 to get -17550.

$$1050x-10x^{2}-17550=0$$

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.

$$-10x^{2}+1050x-17550=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 1050 for b, and -17550 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1050±\sqrt{1050^{2}-4\left(-10\right)\left(-17550\right)}}{2\left(-10\right)}$$

Square 1050.

$$x=\frac{-1050±\sqrt{1102500-4\left(-10\right)\left(-17550\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-1050±\sqrt{1102500+40\left(-17550\right)}}{2\left(-10\right)}$$

Multiply 40 times -17550.

$$x=\frac{-1050±\sqrt{1102500-702000}}{2\left(-10\right)}$$

Add 1102500 to -702000.

$$x=\frac{-1050±\sqrt{400500}}{2\left(-10\right)}$$

Take the square root of 400500.

$$x=\frac{-1050±30\sqrt{445}}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-1050±30\sqrt{445}}{-20}$$

Now solve the equation x=\frac{-1050±30\sqrt{445}}{-20} when ± is plus. Add -1050 to 30\sqrt{445}.

$$x=\frac{30\sqrt{445}-1050}{-20}$$

Divide -1050+30\sqrt{445} by -20.

$$x=\frac{105-3\sqrt{445}}{2}$$

Now solve the equation x=\frac{-1050±30\sqrt{445}}{-20} when ± is minus. Subtract 30\sqrt{445} from -1050.

$$x=\frac{-30\sqrt{445}-1050}{-20}$$

Divide -1050-30\sqrt{445} by -20.

$$x=\frac{3\sqrt{445}+105}{2}$$

The equation is now solved.

$$x=\frac{105-3\sqrt{445}}{2} x=\frac{3\sqrt{445}+105}{2}$$

Steps for Completing the Square

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

$$\left(x-20\right)\left(600-10x+250\right)=550$$

Add 600 and 250 to get 850.

$$\left(x-20\right)\left(850-10x\right)=550$$

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

$$850x-10x^{2}-17000+200x=550$$

Combine 850x and 200x to get 1050x.

$$1050x-10x^{2}-17000=550$$

Add 17000 to both sides.

$$1050x-10x^{2}=550+17000$$

Add 550 and 17000 to get 17550.

$$1050x-10x^{2}=17550$$

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.

$$-10x^{2}+1050x=17550$$

Divide both sides by -10.

$$\frac{-10x^{2}+1050x}{-10}=\frac{17550}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{1050}{-10}x=\frac{17550}{-10}$$

Divide 1050 by -10.

$$x^{2}-105x=\frac{17550}{-10}$$

Divide 17550 by -10.

$$x^{2}-105x=-1755$$

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.

$$x^{2}-105x+\left(-\frac{105}{2}\right)^{2}=-1755+\left(-\frac{105}{2}\right)^{2}$$

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

$$x^{2}-105x+\frac{11025}{4}=-1755+\frac{11025}{4}$$

Add -1755 to \frac{11025}{4}.

$$x^{2}-105x+\frac{11025}{4}=\frac{4005}{4}$$

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}.

$$\left(x-\frac{105}{2}\right)^{2}=\frac{4005}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{105}{2}\right)^{2}}=\sqrt{\frac{4005}{4}}$$

Simplify.

$$x-\frac{105}{2}=\frac{3\sqrt{445}}{2} x-\frac{105}{2}=-\frac{3\sqrt{445}}{2}$$

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

$$x=\frac{3\sqrt{445}+105}{2} x=\frac{105-3\sqrt{445}}{2}$$