Solve -5x^2+15000x=1250000

$$-5 { x }^{ 2 } +15000x = 1250000$$

Solve for x

$$x=1000\sqrt{2}+1500\approx 2914.213562373$$

$$x=1500-1000\sqrt{2}\approx 85.786437627$$

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.

$$-5x^{2}+15000x=1250000$$

Subtract 1250000 from both sides of the equation.

$$-5x^{2}+15000x-1250000=1250000-1250000$$

Subtracting 1250000 from itself leaves 0.

$$-5x^{2}+15000x-1250000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 15000 for b, and -1250000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15000±\sqrt{15000^{2}-4\left(-5\right)\left(-1250000\right)}}{2\left(-5\right)}$$

Square 15000.

$$x=\frac{-15000±\sqrt{225000000-4\left(-5\right)\left(-1250000\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-15000±\sqrt{225000000+20\left(-1250000\right)}}{2\left(-5\right)}$$

Multiply 20 times -1250000.

$$x=\frac{-15000±\sqrt{225000000-25000000}}{2\left(-5\right)}$$

Add 225000000 to -25000000.

$$x=\frac{-15000±\sqrt{200000000}}{2\left(-5\right)}$$

Take the square root of 200000000.

$$x=\frac{-15000±10000\sqrt{2}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-15000±10000\sqrt{2}}{-10}$$

Now solve the equation x=\frac{-15000±10000\sqrt{2}}{-10} when ± is plus. Add -15000 to 10000\sqrt{2}.

$$x=\frac{10000\sqrt{2}-15000}{-10}$$

Divide -15000+10000\sqrt{2} by -10.

$$x=1500-1000\sqrt{2}$$

Now solve the equation x=\frac{-15000±10000\sqrt{2}}{-10} when ± is minus. Subtract 10000\sqrt{2} from -15000.

$$x=\frac{-10000\sqrt{2}-15000}{-10}$$

Divide -15000-10000\sqrt{2} by -10.

$$x=1000\sqrt{2}+1500$$

The equation is now solved.

$$x=1500-1000\sqrt{2} x=1000\sqrt{2}+1500$$

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.

$$-5x^{2}+15000x=1250000$$

Divide both sides by -5.

$$\frac{-5x^{2}+15000x}{-5}=\frac{1250000}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{15000}{-5}x=\frac{1250000}{-5}$$

Divide 15000 by -5.

$$x^{2}-3000x=\frac{1250000}{-5}$$

Divide 1250000 by -5.

$$x^{2}-3000x=-250000$$

Divide -3000, the coefficient of the x term, by 2 to get -1500. Then add the square of -1500 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3000x+\left(-1500\right)^{2}=-250000+\left(-1500\right)^{2}$$

Square -1500.

$$x^{2}-3000x+2250000=-250000+2250000$$

Add -250000 to 2250000.

$$x^{2}-3000x+2250000=2000000$$

Factor x^{2}-3000x+2250000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1500\right)^{2}=2000000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1500\right)^{2}}=\sqrt{2000000}$$

Simplify.

$$x-1500=1000\sqrt{2} x-1500=-1000\sqrt{2}$$

Add 1500 to both sides of the equation.

$$x=1000\sqrt{2}+1500 x=1500-1000\sqrt{2}$$