Solve (100-x)(300+5x)=3200

$$( 100 - x ) ( 300 + 5 x ) = 3200$$

Solve for x

$$x=24\sqrt{10}+20\approx 95.894663844$$

$$x=20-24\sqrt{10}\approx -55.894663844$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 100-x by 300+5x and combine like terms.

$$30000+200x-5x^{2}=3200$$

Subtract 3200 from both sides.

$$30000+200x-5x^{2}-3200=0$$

Subtract 3200 from 30000 to get 26800.

$$26800+200x-5x^{2}=0$$

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.

$$-5x^{2}+200x+26800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 200 for b, and 26800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-200±\sqrt{200^{2}-4\left(-5\right)\times 26800}}{2\left(-5\right)}$$

Square 200.

$$x=\frac{-200±\sqrt{40000-4\left(-5\right)\times 26800}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-200±\sqrt{40000+20\times 26800}}{2\left(-5\right)}$$

Multiply 20 times 26800.

$$x=\frac{-200±\sqrt{40000+536000}}{2\left(-5\right)}$$

Add 40000 to 536000.

$$x=\frac{-200±\sqrt{576000}}{2\left(-5\right)}$$

Take the square root of 576000.

$$x=\frac{-200±240\sqrt{10}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-200±240\sqrt{10}}{-10}$$

Now solve the equation x=\frac{-200±240\sqrt{10}}{-10} when ± is plus. Add -200 to 240\sqrt{10}.

$$x=\frac{240\sqrt{10}-200}{-10}$$

Divide -200+240\sqrt{10} by -10.

$$x=20-24\sqrt{10}$$

Now solve the equation x=\frac{-200±240\sqrt{10}}{-10} when ± is minus. Subtract 240\sqrt{10} from -200.

$$x=\frac{-240\sqrt{10}-200}{-10}$$

Divide -200-240\sqrt{10} by -10.

$$x=24\sqrt{10}+20$$

The equation is now solved.

$$x=20-24\sqrt{10} x=24\sqrt{10}+20$$

Steps for Completing the Square

Use the distributive property to multiply 100-x by 300+5x and combine like terms.

$$30000+200x-5x^{2}=3200$$

Subtract 30000 from both sides.

$$200x-5x^{2}=3200-30000$$

Subtract 30000 from 3200 to get -26800.

$$200x-5x^{2}=-26800$$

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.

$$-5x^{2}+200x=-26800$$

Divide both sides by -5.

$$\frac{-5x^{2}+200x}{-5}=-\frac{26800}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{200}{-5}x=-\frac{26800}{-5}$$

Divide 200 by -5.

$$x^{2}-40x=-\frac{26800}{-5}$$

Divide -26800 by -5.

$$x^{2}-40x=5360$$

Divide -40, the coefficient of the x term, by 2 to get -20. Then add the square of -20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-40x+\left(-20\right)^{2}=5360+\left(-20\right)^{2}$$

Square -20.

$$x^{2}-40x+400=5360+400$$

Add 5360 to 400.

$$x^{2}-40x+400=5760$$

Factor x^{2}-40x+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-20\right)^{2}=5760$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-20\right)^{2}}=\sqrt{5760}$$

Simplify.

$$x-20=24\sqrt{10} x-20=-24\sqrt{10}$$

Add 20 to both sides of the equation.

$$x=24\sqrt{10}+20 x=20-24\sqrt{10}$$