Solve -5x^2+200x+30000=3200

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

Solve for x

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

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

Steps Using the Quadratic Formula

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.

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

Subtract 3200 from both sides of the equation.

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

Subtracting 3200 from itself leaves 0.

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

Subtract 3200 from 30000.

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

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

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.

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

Subtract 30000 from both sides of the equation.

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

Subtracting 30000 from itself leaves 0.

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

Subtract 30000 from 3200.

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

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.

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