Solve x^2+600x=18000

$$x ^ { 2 } + 600 x = 18000$$

Solve for x

$$x=60\sqrt{30}-300\approx 28.633534503$$

$$x=-60\sqrt{30}-300\approx -628.633534503$$

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.

$$x^{2}+600x=18000$$

Subtract 18000 from both sides of the equation.

$$x^{2}+600x-18000=18000-18000$$

Subtracting 18000 from itself leaves 0.

$$x^{2}+600x-18000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 600 for b, and -18000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-600±\sqrt{600^{2}-4\left(-18000\right)}}{2}$$

Square 600.

$$x=\frac{-600±\sqrt{360000-4\left(-18000\right)}}{2}$$

Multiply -4 times -18000.

$$x=\frac{-600±\sqrt{360000+72000}}{2}$$

Add 360000 to 72000.

$$x=\frac{-600±\sqrt{432000}}{2}$$

Take the square root of 432000.

$$x=\frac{-600±120\sqrt{30}}{2}$$

Now solve the equation x=\frac{-600±120\sqrt{30}}{2} when ± is plus. Add -600 to 120\sqrt{30}.

$$x=\frac{120\sqrt{30}-600}{2}$$

Divide -600+120\sqrt{30} by 2.

$$x=60\sqrt{30}-300$$

Now solve the equation x=\frac{-600±120\sqrt{30}}{2} when ± is minus. Subtract 120\sqrt{30} from -600.

$$x=\frac{-120\sqrt{30}-600}{2}$$

Divide -600-120\sqrt{30} by 2.

$$x=-60\sqrt{30}-300$$

The equation is now solved.

$$x=60\sqrt{30}-300 x=-60\sqrt{30}-300$$

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.

$$x^{2}+600x=18000$$

Divide 600, the coefficient of the x term, by 2 to get 300. Then add the square of 300 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+600x+300^{2}=18000+300^{2}$$

Square 300.

$$x^{2}+600x+90000=18000+90000$$

Add 18000 to 90000.

$$x^{2}+600x+90000=108000$$

Factor x^{2}+600x+90000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+300\right)^{2}=108000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+300\right)^{2}}=\sqrt{108000}$$

Simplify.

$$x+300=60\sqrt{30} x+300=-60\sqrt{30}$$

Subtract 300 from both sides of the equation.

$$x=60\sqrt{30}-300 x=-60\sqrt{30}-300$$