Solve (x-100)[300+5(200-x)]=3200

$$( x - 100 ) [ 300 + 5 ( 200 - x ) ] = 3200$$

Solve for x

$$x=24\sqrt{10}+180\approx 255.894663844$$

$$x=180-24\sqrt{10}\approx 104.105336156$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5 by 200-x.

$$\left(x-100\right)\left(300+1000-5x\right)=3200$$

Add 300 and 1000 to get 1300.

$$\left(x-100\right)\left(1300-5x\right)=3200$$

Apply the distributive property by multiplying each term of x-100 by each term of 1300-5x.

$$1300x-5x^{2}-130000+500x=3200$$

Combine 1300x and 500x to get 1800x.

$$1800x-5x^{2}-130000=3200$$

Subtract 3200 from both sides.

$$1800x-5x^{2}-130000-3200=0$$

Subtract 3200 from -130000 to get -133200.

$$1800x-5x^{2}-133200=0$$

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.

$$-5x^{2}+1800x-133200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 1800 for b, and -133200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1800±\sqrt{1800^{2}-4\left(-5\right)\left(-133200\right)}}{2\left(-5\right)}$$

Square 1800.

$$x=\frac{-1800±\sqrt{3240000-4\left(-5\right)\left(-133200\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-1800±\sqrt{3240000+20\left(-133200\right)}}{2\left(-5\right)}$$

Multiply 20 times -133200.

$$x=\frac{-1800±\sqrt{3240000-2664000}}{2\left(-5\right)}$$

Add 3240000 to -2664000.

$$x=\frac{-1800±\sqrt{576000}}{2\left(-5\right)}$$

Take the square root of 576000.

$$x=\frac{-1800±240\sqrt{10}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-1800±240\sqrt{10}}{-10}$$

Now solve the equation x=\frac{-1800±240\sqrt{10}}{-10} when ± is plus. Add -1800 to 240\sqrt{10}.

$$x=\frac{240\sqrt{10}-1800}{-10}$$

Divide -1800+240\sqrt{10} by -10.

$$x=180-24\sqrt{10}$$

Now solve the equation x=\frac{-1800±240\sqrt{10}}{-10} when ± is minus. Subtract 240\sqrt{10} from -1800.

$$x=\frac{-240\sqrt{10}-1800}{-10}$$

Divide -1800-240\sqrt{10} by -10.

$$x=24\sqrt{10}+180$$

The equation is now solved.

$$x=180-24\sqrt{10} x=24\sqrt{10}+180$$

Steps for Completing the Square

Use the distributive property to multiply 5 by 200-x.

$$\left(x-100\right)\left(300+1000-5x\right)=3200$$

Add 300 and 1000 to get 1300.

$$\left(x-100\right)\left(1300-5x\right)=3200$$

Apply the distributive property by multiplying each term of x-100 by each term of 1300-5x.

$$1300x-5x^{2}-130000+500x=3200$$

Combine 1300x and 500x to get 1800x.

$$1800x-5x^{2}-130000=3200$$

Add 130000 to both sides.

$$1800x-5x^{2}=3200+130000$$

Add 3200 and 130000 to get 133200.

$$1800x-5x^{2}=133200$$

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.

$$-5x^{2}+1800x=133200$$

Divide both sides by -5.

$$\frac{-5x^{2}+1800x}{-5}=\frac{133200}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{1800}{-5}x=\frac{133200}{-5}$$

Divide 1800 by -5.

$$x^{2}-360x=\frac{133200}{-5}$$

Divide 133200 by -5.

$$x^{2}-360x=-26640$$

Divide -360, the coefficient of the x term, by 2 to get -180. Then add the square of -180 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-360x+\left(-180\right)^{2}=-26640+\left(-180\right)^{2}$$

Square -180.

$$x^{2}-360x+32400=-26640+32400$$

Add -26640 to 32400.

$$x^{2}-360x+32400=5760$$

Factor x^{2}-360x+32400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-180\right)^{2}=5760$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-180\right)^{2}}=\sqrt{5760}$$

Simplify.

$$x-180=24\sqrt{10} x-180=-24\sqrt{10}$$

Add 180 to both sides of the equation.

$$x=24\sqrt{10}+180 x=180-24\sqrt{10}$$