Solve x^2-110x-3000=0

$${ x }^{ 2 } -110x-3000=0$$

Solve for x

$$x=5\sqrt{241}+55\approx 132.620873481$$

$$x=55-5\sqrt{241}\approx -22.620873481$$

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.

$$x^{2}-110x-3000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -110 for b, and -3000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-110\right)±\sqrt{\left(-110\right)^{2}-4\left(-3000\right)}}{2}$$

Square -110.

$$x=\frac{-\left(-110\right)±\sqrt{12100-4\left(-3000\right)}}{2}$$

Multiply -4 times -3000.

$$x=\frac{-\left(-110\right)±\sqrt{12100+12000}}{2}$$

Add 12100 to 12000.

$$x=\frac{-\left(-110\right)±\sqrt{24100}}{2}$$

Take the square root of 24100.

$$x=\frac{-\left(-110\right)±10\sqrt{241}}{2}$$

The opposite of -110 is 110.

$$x=\frac{110±10\sqrt{241}}{2}$$

Now solve the equation x=\frac{110±10\sqrt{241}}{2} when ± is plus. Add 110 to 10\sqrt{241}.

$$x=\frac{10\sqrt{241}+110}{2}$$

Divide 110+10\sqrt{241} by 2.

$$x=5\sqrt{241}+55$$

Now solve the equation x=\frac{110±10\sqrt{241}}{2} when ± is minus. Subtract 10\sqrt{241} from 110.

$$x=\frac{110-10\sqrt{241}}{2}$$

Divide 110-10\sqrt{241} by 2.

$$x=55-5\sqrt{241}$$

The equation is now solved.

$$x=5\sqrt{241}+55 x=55-5\sqrt{241}$$

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.

$$x^{2}-110x-3000=0$$

Add 3000 to both sides of the equation.

$$x^{2}-110x-3000-\left(-3000\right)=-\left(-3000\right)$$

Subtracting -3000 from itself leaves 0.

$$x^{2}-110x=-\left(-3000\right)$$

Subtract -3000 from 0.

$$x^{2}-110x=3000$$

Divide -110, the coefficient of the x term, by 2 to get -55. Then add the square of -55 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-110x+\left(-55\right)^{2}=3000+\left(-55\right)^{2}$$

Square -55.

$$x^{2}-110x+3025=3000+3025$$

Add 3000 to 3025.

$$x^{2}-110x+3025=6025$$

Factor x^{2}-110x+3025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-55\right)^{2}=6025$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-55\right)^{2}}=\sqrt{6025}$$

Simplify.

$$x-55=5\sqrt{241} x-55=-5\sqrt{241}$$

Add 55 to both sides of the equation.

$$x=5\sqrt{241}+55 x=55-5\sqrt{241}$$