Solve x(-11x/5)+5(-11x/5)=22

$$x(- \frac{ 11x }{ 5 } )+5(- \frac{ 11x }{ 5 } )=22$$

Solve for x (complex solution)

$$x=\frac{-5+\sqrt{15}i}{2}\approx -2.5+1.936491673i$$

$$x=\frac{-\sqrt{15}i-5}{2}\approx -2.5-1.936491673i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 5.

$$5x\left(-\frac{11x}{5}\right)+25\left(-\frac{11x}{5}\right)=110$$

Express 5\left(-\frac{11x}{5}\right) as a single fraction.

$$\frac{-5\times 11x}{5}x+25\left(-\frac{11x}{5}\right)=110$$

Cancel out 5 and 5.

$$-11xx+25\left(-\frac{11x}{5}\right)=110$$

Cancel out 5, the greatest common factor in 25 and 5.

$$-11xx-5\times 11x=110$$

Multiply -1 and 11 to get -11. Multiply -5 and 11 to get -55.

$$-11xx-55x=110$$

Multiply x and x to get x^{2}.

$$-11x^{2}-55x=110$$

Subtract 110 from both sides.

$$-11x^{2}-55x-110=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -11 for a, -55 for b, and -110 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-55\right)±\sqrt{\left(-55\right)^{2}-4\left(-11\right)\left(-110\right)}}{2\left(-11\right)}$$

Square -55.

$$x=\frac{-\left(-55\right)±\sqrt{3025-4\left(-11\right)\left(-110\right)}}{2\left(-11\right)}$$

Multiply -4 times -11.

$$x=\frac{-\left(-55\right)±\sqrt{3025+44\left(-110\right)}}{2\left(-11\right)}$$

Multiply 44 times -110.

$$x=\frac{-\left(-55\right)±\sqrt{3025-4840}}{2\left(-11\right)}$$

Add 3025 to -4840.

$$x=\frac{-\left(-55\right)±\sqrt{-1815}}{2\left(-11\right)}$$

Take the square root of -1815.

$$x=\frac{-\left(-55\right)±11\sqrt{15}i}{2\left(-11\right)}$$

The opposite of -55 is 55.

$$x=\frac{55±11\sqrt{15}i}{2\left(-11\right)}$$

Multiply 2 times -11.

$$x=\frac{55±11\sqrt{15}i}{-22}$$

Now solve the equation x=\frac{55±11\sqrt{15}i}{-22} when ± is plus. Add 55 to 11i\sqrt{15}.

$$x=\frac{55+11\sqrt{15}i}{-22}$$

Divide 55+11i\sqrt{15} by -22.

$$x=\frac{-\sqrt{15}i-5}{2}$$

Now solve the equation x=\frac{55±11\sqrt{15}i}{-22} when ± is minus. Subtract 11i\sqrt{15} from 55.

$$x=\frac{-11\sqrt{15}i+55}{-22}$$

Divide 55-11i\sqrt{15} by -22.

$$x=\frac{-5+\sqrt{15}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{15}i-5}{2} x=\frac{-5+\sqrt{15}i}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 5.

$$5x\left(-\frac{11x}{5}\right)+25\left(-\frac{11x}{5}\right)=110$$

Express 5\left(-\frac{11x}{5}\right) as a single fraction.

$$\frac{-5\times 11x}{5}x+25\left(-\frac{11x}{5}\right)=110$$

Cancel out 5 and 5.

$$-11xx+25\left(-\frac{11x}{5}\right)=110$$

Cancel out 5, the greatest common factor in 25 and 5.

$$-11xx-5\times 11x=110$$

Multiply -1 and 11 to get -11. Multiply -5 and 11 to get -55.

$$-11xx-55x=110$$

Multiply x and x to get x^{2}.

$$-11x^{2}-55x=110$$

Divide both sides by -11.

$$\frac{-11x^{2}-55x}{-11}=\frac{110}{-11}$$

Dividing by -11 undoes the multiplication by -11.

$$x^{2}+\left(-\frac{55}{-11}\right)x=\frac{110}{-11}$$

Divide -55 by -11.

$$x^{2}+5x=\frac{110}{-11}$$

Divide 110 by -11.

$$x^{2}+5x=-10$$

Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=-10+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+5x+\frac{25}{4}=-10+\frac{25}{4}$$

Add -10 to \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=-\frac{15}{4}$$

Factor x^{2}+5x+\frac{25}{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}.

$$\left(x+\frac{5}{2}\right)^{2}=-\frac{15}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{-\frac{15}{4}}$$

Simplify.

$$x+\frac{5}{2}=\frac{\sqrt{15}i}{2} x+\frac{5}{2}=-\frac{\sqrt{15}i}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=\frac{-5+\sqrt{15}i}{2} x=\frac{-\sqrt{15}i-5}{2}$$