Solve x^2-700x-480000=0

$${ x }^{ 2 } -700x-480000=0$$

Solve for x

$$x=50\sqrt{241}+350\approx 1126.208734813$$

$$x=350-50\sqrt{241}\approx -426.208734813$$

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.

$$x^{2}-700x-480000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -700 for b, and -480000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-700\right)±\sqrt{\left(-700\right)^{2}-4\left(-480000\right)}}{2}$$

Square -700.

$$x=\frac{-\left(-700\right)±\sqrt{490000-4\left(-480000\right)}}{2}$$

Multiply -4 times -480000.

$$x=\frac{-\left(-700\right)±\sqrt{490000+1920000}}{2}$$

Add 490000 to 1920000.

$$x=\frac{-\left(-700\right)±\sqrt{2410000}}{2}$$

Take the square root of 2410000.

$$x=\frac{-\left(-700\right)±100\sqrt{241}}{2}$$

The opposite of -700 is 700.

$$x=\frac{700±100\sqrt{241}}{2}$$

Now solve the equation x=\frac{700±100\sqrt{241}}{2} when ± is plus. Add 700 to 100\sqrt{241}.

$$x=\frac{100\sqrt{241}+700}{2}$$

Divide 700+100\sqrt{241} by 2.

$$x=50\sqrt{241}+350$$

Now solve the equation x=\frac{700±100\sqrt{241}}{2} when ± is minus. Subtract 100\sqrt{241} from 700.

$$x=\frac{700-100\sqrt{241}}{2}$$

Divide 700-100\sqrt{241} by 2.

$$x=350-50\sqrt{241}$$

The equation is now solved.

$$x=50\sqrt{241}+350 x=350-50\sqrt{241}$$

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.

$$x^{2}-700x-480000=0$$

Add 480000 to both sides of the equation.

$$x^{2}-700x-480000-\left(-480000\right)=-\left(-480000\right)$$

Subtracting -480000 from itself leaves 0.

$$x^{2}-700x=-\left(-480000\right)$$

Subtract -480000 from 0.

$$x^{2}-700x=480000$$

Divide -700, the coefficient of the x term, by 2 to get -350. Then add the square of -350 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-700x+\left(-350\right)^{2}=480000+\left(-350\right)^{2}$$

Square -350.

$$x^{2}-700x+122500=480000+122500$$

Add 480000 to 122500.

$$x^{2}-700x+122500=602500$$

Factor x^{2}-700x+122500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-350\right)^{2}=602500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-350\right)^{2}}=\sqrt{602500}$$

Simplify.

$$x-350=50\sqrt{241} x-350=-50\sqrt{241}$$

Add 350 to both sides of the equation.

$$x=50\sqrt{241}+350 x=350-50\sqrt{241}$$