Solve x^2+100x-500=0

$${ x }^{ 2 } +100x-500=0$$

Solve for x

$$x=10\sqrt{30}-50\approx 4.772255751$$

$$x=-10\sqrt{30}-50\approx -104.772255751$$

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.

$$x^{2}+100x-500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 100 for b, and -500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-500\right)}}{2}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-500\right)}}{2}$$

Multiply -4 times -500.

$$x=\frac{-100±\sqrt{10000+2000}}{2}$$

Add 10000 to 2000.

$$x=\frac{-100±\sqrt{12000}}{2}$$

Take the square root of 12000.

$$x=\frac{-100±20\sqrt{30}}{2}$$

Now solve the equation x=\frac{-100±20\sqrt{30}}{2} when ± is plus. Add -100 to 20\sqrt{30}.

$$x=\frac{20\sqrt{30}-100}{2}$$

Divide -100+20\sqrt{30} by 2.

$$x=10\sqrt{30}-50$$

Now solve the equation x=\frac{-100±20\sqrt{30}}{2} when ± is minus. Subtract 20\sqrt{30} from -100.

$$x=\frac{-20\sqrt{30}-100}{2}$$

Divide -100-20\sqrt{30} by 2.

$$x=-10\sqrt{30}-50$$

The equation is now solved.

$$x=10\sqrt{30}-50 x=-10\sqrt{30}-50$$

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.

$$x^{2}+100x-500=0$$

Add 500 to both sides of the equation.

$$x^{2}+100x-500-\left(-500\right)=-\left(-500\right)$$

Subtracting -500 from itself leaves 0.

$$x^{2}+100x=-\left(-500\right)$$

Subtract -500 from 0.

$$x^{2}+100x=500$$

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.

$$x^{2}+100x+50^{2}=500+50^{2}$$

Square 50.

$$x^{2}+100x+2500=500+2500$$

Add 500 to 2500.

$$x^{2}+100x+2500=3000$$

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}.

$$\left(x+50\right)^{2}=3000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+50\right)^{2}}=\sqrt{3000}$$

Simplify.

$$x+50=10\sqrt{30} x+50=-10\sqrt{30}$$

Subtract 50 from both sides of the equation.

$$x=10\sqrt{30}-50 x=-10\sqrt{30}-50$$