Solve x^2+250x-12500=0

$${ x }^{ 2 } +250x-12500=0$$

Solve for x

$$x=75\sqrt{5}-125\approx 42.705098312$$

$$x=-75\sqrt{5}-125\approx -292.705098312$$

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.

$$x^{2}+250x-12500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 250 for b, and -12500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-250±\sqrt{250^{2}-4\left(-12500\right)}}{2}$$

Square 250.

$$x=\frac{-250±\sqrt{62500-4\left(-12500\right)}}{2}$$

Multiply -4 times -12500.

$$x=\frac{-250±\sqrt{62500+50000}}{2}$$

Add 62500 to 50000.

$$x=\frac{-250±\sqrt{112500}}{2}$$

Take the square root of 112500.

$$x=\frac{-250±150\sqrt{5}}{2}$$

Now solve the equation x=\frac{-250±150\sqrt{5}}{2} when ± is plus. Add -250 to 150\sqrt{5}.

$$x=\frac{150\sqrt{5}-250}{2}$$

Divide -250+150\sqrt{5} by 2.

$$x=75\sqrt{5}-125$$

Now solve the equation x=\frac{-250±150\sqrt{5}}{2} when ± is minus. Subtract 150\sqrt{5} from -250.

$$x=\frac{-150\sqrt{5}-250}{2}$$

Divide -250-150\sqrt{5} by 2.

$$x=-75\sqrt{5}-125$$

The equation is now solved.

$$x=75\sqrt{5}-125 x=-75\sqrt{5}-125$$

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.

$$x^{2}+250x-12500=0$$

Add 12500 to both sides of the equation.

$$x^{2}+250x-12500-\left(-12500\right)=-\left(-12500\right)$$

Subtracting -12500 from itself leaves 0.

$$x^{2}+250x=-\left(-12500\right)$$

Subtract -12500 from 0.

$$x^{2}+250x=12500$$

Divide 250, the coefficient of the x term, by 2 to get 125. Then add the square of 125 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+250x+125^{2}=12500+125^{2}$$

Square 125.

$$x^{2}+250x+15625=12500+15625$$

Add 12500 to 15625.

$$x^{2}+250x+15625=28125$$

Factor x^{2}+250x+15625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+125\right)^{2}=28125$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+125\right)^{2}}=\sqrt{28125}$$

Simplify.

$$x+125=75\sqrt{5} x+125=-75\sqrt{5}$$

Subtract 125 from both sides of the equation.

$$x=75\sqrt{5}-125 x=-75\sqrt{5}-125$$