Solve -10x^2+500x+6000=10000

$$- 10 x ^ { 2 } + 500 x + 6000 = 10000$$

Solve for x

$$x=10$$

$$x=40$$

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.

$$-10x^{2}+500x+6000=10000$$

Subtract 10000 from both sides of the equation.

$$-10x^{2}+500x+6000-10000=10000-10000$$

Subtracting 10000 from itself leaves 0.

$$-10x^{2}+500x+6000-10000=0$$

Subtract 10000 from 6000.

$$-10x^{2}+500x-4000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 500 for b, and -4000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-500±\sqrt{500^{2}-4\left(-10\right)\left(-4000\right)}}{2\left(-10\right)}$$

Square 500.

$$x=\frac{-500±\sqrt{250000-4\left(-10\right)\left(-4000\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-500±\sqrt{250000+40\left(-4000\right)}}{2\left(-10\right)}$$

Multiply 40 times -4000.

$$x=\frac{-500±\sqrt{250000-160000}}{2\left(-10\right)}$$

Add 250000 to -160000.

$$x=\frac{-500±\sqrt{90000}}{2\left(-10\right)}$$

Take the square root of 90000.

$$x=\frac{-500±300}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-500±300}{-20}$$

Now solve the equation x=\frac{-500±300}{-20} when ± is plus. Add -500 to 300.

$$x=-\frac{200}{-20}$$

Divide -200 by -20.

$$x=10$$

Now solve the equation x=\frac{-500±300}{-20} when ± is minus. Subtract 300 from -500.

$$x=-\frac{800}{-20}$$

Divide -800 by -20.

$$x=40$$

The equation is now solved.

$$x=10 x=40$$

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.

$$-10x^{2}+500x+6000=10000$$

Subtract 6000 from both sides of the equation.

$$-10x^{2}+500x+6000-6000=10000-6000$$

Subtracting 6000 from itself leaves 0.

$$-10x^{2}+500x=10000-6000$$

Subtract 6000 from 10000.

$$-10x^{2}+500x=4000$$

Divide both sides by -10.

$$\frac{-10x^{2}+500x}{-10}=\frac{4000}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{500}{-10}x=\frac{4000}{-10}$$

Divide 500 by -10.

$$x^{2}-50x=\frac{4000}{-10}$$

Divide 4000 by -10.

$$x^{2}-50x=-400$$

Divide -50, the coefficient of the x term, by 2 to get -25. Then add the square of -25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-50x+\left(-25\right)^{2}=-400+\left(-25\right)^{2}$$

Square -25.

$$x^{2}-50x+625=-400+625$$

Add -400 to 625.

$$x^{2}-50x+625=225$$

Factor x^{2}-50x+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-25\right)^{2}=225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{225}$$

Simplify.

$$x-25=15 x-25=-15$$

Add 25 to both sides of the equation.

$$x=40 x=10$$