Solve 2.5x^2+250x-15000=0

$$2.5 { x }^{ 2 } +250x-15000=0$$

Solve for x

$$x=10\sqrt{85}-50\approx 42.195444573$$

$$x=-10\sqrt{85}-50\approx -142.195444573$$

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.

$$2.5x^{2}+250x-15000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2.5 for a, 250 for b, and -15000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-250±\sqrt{250^{2}-4\times 2.5\left(-15000\right)}}{2\times 2.5}$$

Square 250.

$$x=\frac{-250±\sqrt{62500-4\times 2.5\left(-15000\right)}}{2\times 2.5}$$

Multiply -4 times 2.5.

$$x=\frac{-250±\sqrt{62500-10\left(-15000\right)}}{2\times 2.5}$$

Multiply -10 times -15000.

$$x=\frac{-250±\sqrt{62500+150000}}{2\times 2.5}$$

Add 62500 to 150000.

$$x=\frac{-250±\sqrt{212500}}{2\times 2.5}$$

Take the square root of 212500.

$$x=\frac{-250±50\sqrt{85}}{2\times 2.5}$$

Multiply 2 times 2.5.

$$x=\frac{-250±50\sqrt{85}}{5}$$

Now solve the equation x=\frac{-250±50\sqrt{85}}{5} when ± is plus. Add -250 to 50\sqrt{85}.

$$x=\frac{50\sqrt{85}-250}{5}$$

Divide -250+50\sqrt{85} by 5.

$$x=10\sqrt{85}-50$$

Now solve the equation x=\frac{-250±50\sqrt{85}}{5} when ± is minus. Subtract 50\sqrt{85} from -250.

$$x=\frac{-50\sqrt{85}-250}{5}$$

Divide -250-50\sqrt{85} by 5.

$$x=-10\sqrt{85}-50$$

The equation is now solved.

$$x=10\sqrt{85}-50 x=-10\sqrt{85}-50$$

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.

$$2.5x^{2}+250x-15000=0$$

Add 15000 to both sides of the equation.

$$2.5x^{2}+250x-15000-\left(-15000\right)=-\left(-15000\right)$$

Subtracting -15000 from itself leaves 0.

$$2.5x^{2}+250x=-\left(-15000\right)$$

Subtract -15000 from 0.

$$2.5x^{2}+250x=15000$$

Divide both sides of the equation by 2.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{2.5x^{2}+250x}{2.5}=\frac{15000}{2.5}$$

Dividing by 2.5 undoes the multiplication by 2.5.

$$x^{2}+\frac{250}{2.5}x=\frac{15000}{2.5}$$

Divide 250 by 2.5 by multiplying 250 by the reciprocal of 2.5.

$$x^{2}+100x=\frac{15000}{2.5}$$

Divide 15000 by 2.5 by multiplying 15000 by the reciprocal of 2.5.

$$x^{2}+100x=6000$$

Divide 100, the coefficient of the x term, by 2 to get 50. Then add the square of 50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+100x+50^{2}=6000+50^{2}$$

Square 50.

$$x^{2}+100x+2500=6000+2500$$

Add 6000 to 2500.

$$x^{2}+100x+2500=8500$$

Factor x^{2}+100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+50\right)^{2}=8500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+50\right)^{2}}=\sqrt{8500}$$

Simplify.

$$x+50=10\sqrt{85} x+50=-10\sqrt{85}$$

Subtract 50 from both sides of the equation.

$$x=10\sqrt{85}-50 x=-10\sqrt{85}-50$$