Solve 11x^2+106x+259=0

$$11 { x }^{ 2 } +106x+259=0$$

Solve for x (complex solution)

$$x=\frac{-53+2\sqrt{10}i}{11}\approx -4.818181818+0.574959575i$$

$$x=\frac{-2\sqrt{10}i-53}{11}\approx -4.818181818-0.574959575i$$

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.

$$11x^{2}+106x+259=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 11 for a, 106 for b, and 259 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-106±\sqrt{106^{2}-4\times 11\times 259}}{2\times 11}$$

Square 106.

$$x=\frac{-106±\sqrt{11236-4\times 11\times 259}}{2\times 11}$$

Multiply -4 times 11.

$$x=\frac{-106±\sqrt{11236-44\times 259}}{2\times 11}$$

Multiply -44 times 259.

$$x=\frac{-106±\sqrt{11236-11396}}{2\times 11}$$

Add 11236 to -11396.

$$x=\frac{-106±\sqrt{-160}}{2\times 11}$$

Take the square root of -160.

$$x=\frac{-106±4\sqrt{10}i}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{-106±4\sqrt{10}i}{22}$$

Now solve the equation x=\frac{-106±4\sqrt{10}i}{22} when ± is plus. Add -106 to 4i\sqrt{10}.

$$x=\frac{-106+4\sqrt{10}i}{22}$$

Divide -106+4i\sqrt{10} by 22.

$$x=\frac{-53+2\sqrt{10}i}{11}$$

Now solve the equation x=\frac{-106±4\sqrt{10}i}{22} when ± is minus. Subtract 4i\sqrt{10} from -106.

$$x=\frac{-4\sqrt{10}i-106}{22}$$

Divide -106-4i\sqrt{10} by 22.

$$x=\frac{-2\sqrt{10}i-53}{11}$$

The equation is now solved.

$$x=\frac{-53+2\sqrt{10}i}{11} x=\frac{-2\sqrt{10}i-53}{11}$$

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.

$$11x^{2}+106x+259=0$$

Subtract 259 from both sides of the equation.

$$11x^{2}+106x+259-259=-259$$

Subtracting 259 from itself leaves 0.

$$11x^{2}+106x=-259$$

Divide both sides by 11.

$$\frac{11x^{2}+106x}{11}=-\frac{259}{11}$$

Dividing by 11 undoes the multiplication by 11.

$$x^{2}+\frac{106}{11}x=-\frac{259}{11}$$

Divide \frac{106}{11}, the coefficient of the x term, by 2 to get \frac{53}{11}. Then add the square of \frac{53}{11} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{106}{11}x+\left(\frac{53}{11}\right)^{2}=-\frac{259}{11}+\left(\frac{53}{11}\right)^{2}$$

Square \frac{53}{11} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{106}{11}x+\frac{2809}{121}=-\frac{259}{11}+\frac{2809}{121}$$

Add -\frac{259}{11} to \frac{2809}{121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{106}{11}x+\frac{2809}{121}=-\frac{40}{121}$$

Factor x^{2}+\frac{106}{11}x+\frac{2809}{121}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{53}{11}\right)^{2}=-\frac{40}{121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{53}{11}\right)^{2}}=\sqrt{-\frac{40}{121}}$$

Simplify.

$$x+\frac{53}{11}=\frac{2\sqrt{10}i}{11} x+\frac{53}{11}=-\frac{2\sqrt{10}i}{11}$$

Subtract \frac{53}{11} from both sides of the equation.

$$x=\frac{-53+2\sqrt{10}i}{11} x=\frac{-2\sqrt{10}i-53}{11}$$