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

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

Solve for x

$$x=40\sqrt{23}+300\approx 491.833260933$$

$$x=300-40\sqrt{23}\approx 108.166739067$$

Steps Using the Quadratic Formula

Add 300 and 200 to get 500.

$$\left(x-100\right)\left(500-x\right)=3200$$

Use the distributive property to multiply x-100 by 500-x and combine like terms.

$$600x-x^{2}-50000=3200$$

Subtract 3200 from both sides.

$$600x-x^{2}-50000-3200=0$$

Subtract 3200 from -50000 to get -53200.

$$600x-x^{2}-53200=0$$

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.

$$-x^{2}+600x-53200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 600 for b, and -53200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-600±\sqrt{600^{2}-4\left(-1\right)\left(-53200\right)}}{2\left(-1\right)}$$

Square 600.

$$x=\frac{-600±\sqrt{360000-4\left(-1\right)\left(-53200\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-600±\sqrt{360000+4\left(-53200\right)}}{2\left(-1\right)}$$

Multiply 4 times -53200.

$$x=\frac{-600±\sqrt{360000-212800}}{2\left(-1\right)}$$

Add 360000 to -212800.

$$x=\frac{-600±\sqrt{147200}}{2\left(-1\right)}$$

Take the square root of 147200.

$$x=\frac{-600±80\sqrt{23}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-600±80\sqrt{23}}{-2}$$

Now solve the equation x=\frac{-600±80\sqrt{23}}{-2} when ± is plus. Add -600 to 80\sqrt{23}.

$$x=\frac{80\sqrt{23}-600}{-2}$$

Divide -600+80\sqrt{23} by -2.

$$x=300-40\sqrt{23}$$

Now solve the equation x=\frac{-600±80\sqrt{23}}{-2} when ± is minus. Subtract 80\sqrt{23} from -600.

$$x=\frac{-80\sqrt{23}-600}{-2}$$

Divide -600-80\sqrt{23} by -2.

$$x=40\sqrt{23}+300$$

The equation is now solved.

$$x=300-40\sqrt{23} x=40\sqrt{23}+300$$

Steps for Completing the Square

Add 300 and 200 to get 500.

$$\left(x-100\right)\left(500-x\right)=3200$$

Use the distributive property to multiply x-100 by 500-x and combine like terms.

$$600x-x^{2}-50000=3200$$

Add 50000 to both sides.

$$600x-x^{2}=3200+50000$$

Add 3200 and 50000 to get 53200.

$$600x-x^{2}=53200$$

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.

$$-x^{2}+600x=53200$$

Divide both sides by -1.

$$\frac{-x^{2}+600x}{-1}=\frac{53200}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{600}{-1}x=\frac{53200}{-1}$$

Divide 600 by -1.

$$x^{2}-600x=\frac{53200}{-1}$$

Divide 53200 by -1.

$$x^{2}-600x=-53200$$

Divide -600, the coefficient of the x term, by 2 to get -300. Then add the square of -300 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-600x+\left(-300\right)^{2}=-53200+\left(-300\right)^{2}$$

Square -300.

$$x^{2}-600x+90000=-53200+90000$$

Add -53200 to 90000.

$$x^{2}-600x+90000=36800$$

Factor x^{2}-600x+90000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-300\right)^{2}=36800$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-300\right)^{2}}=\sqrt{36800}$$

Simplify.

$$x-300=40\sqrt{23} x-300=-40\sqrt{23}$$

Add 300 to both sides of the equation.

$$x=40\sqrt{23}+300 x=300-40\sqrt{23}$$