Solve 110x^2-110x-200=0

$$110 { x }^{ 2 } -110x-200=0$$

Solve for x

$$x=\frac{\sqrt{1001}}{22}+\frac{1}{2}\approx 1.938117456$$

$$x=-\frac{\sqrt{1001}}{22}+\frac{1}{2}\approx -0.938117456$$

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.

$$110x^{2}-110x-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 110 for a, -110 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-110\right)±\sqrt{\left(-110\right)^{2}-4\times 110\left(-200\right)}}{2\times 110}$$

Square -110.

$$x=\frac{-\left(-110\right)±\sqrt{12100-4\times 110\left(-200\right)}}{2\times 110}$$

Multiply -4 times 110.

$$x=\frac{-\left(-110\right)±\sqrt{12100-440\left(-200\right)}}{2\times 110}$$

Multiply -440 times -200.

$$x=\frac{-\left(-110\right)±\sqrt{12100+88000}}{2\times 110}$$

Add 12100 to 88000.

$$x=\frac{-\left(-110\right)±\sqrt{100100}}{2\times 110}$$

Take the square root of 100100.

$$x=\frac{-\left(-110\right)±10\sqrt{1001}}{2\times 110}$$

The opposite of -110 is 110.

$$x=\frac{110±10\sqrt{1001}}{2\times 110}$$

Multiply 2 times 110.

$$x=\frac{110±10\sqrt{1001}}{220}$$

Now solve the equation x=\frac{110±10\sqrt{1001}}{220} when ± is plus. Add 110 to 10\sqrt{1001}.

$$x=\frac{10\sqrt{1001}+110}{220}$$

Divide 110+10\sqrt{1001} by 220.

$$x=\frac{\sqrt{1001}}{22}+\frac{1}{2}$$

Now solve the equation x=\frac{110±10\sqrt{1001}}{220} when ± is minus. Subtract 10\sqrt{1001} from 110.

$$x=\frac{110-10\sqrt{1001}}{220}$$

Divide 110-10\sqrt{1001} by 220.

$$x=-\frac{\sqrt{1001}}{22}+\frac{1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{1001}}{22}+\frac{1}{2} x=-\frac{\sqrt{1001}}{22}+\frac{1}{2}$$

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.

$$110x^{2}-110x-200=0$$

Add 200 to both sides of the equation.

$$110x^{2}-110x-200-\left(-200\right)=-\left(-200\right)$$

Subtracting -200 from itself leaves 0.

$$110x^{2}-110x=-\left(-200\right)$$

Subtract -200 from 0.

$$110x^{2}-110x=200$$

Divide both sides by 110.

$$\frac{110x^{2}-110x}{110}=\frac{200}{110}$$

Dividing by 110 undoes the multiplication by 110.

$$x^{2}+\left(-\frac{110}{110}\right)x=\frac{200}{110}$$

Divide -110 by 110.

$$x^{2}-x=\frac{200}{110}$$

Reduce the fraction \frac{200}{110} to lowest terms by extracting and canceling out 10.

$$x^{2}-x=\frac{20}{11}$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{20}{11}+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=\frac{20}{11}+\frac{1}{4}$$

Add \frac{20}{11} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=\frac{91}{44}$$

Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{91}{44}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{91}{44}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{1001}}{22} x-\frac{1}{2}=-\frac{\sqrt{1001}}{22}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{1001}}{22}+\frac{1}{2} x=-\frac{\sqrt{1001}}{22}+\frac{1}{2}$$