Solve -2x^2+9x-9-4x^2+2x+6-6x^2-12x-18=0

$$-2 { x }^{ 2 } +9x-9-4 { x }^{ 2 } +2x+6-6 { x }^{ 2 } -12x-18=0$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{1007}i-1}{24}\approx -0.041666667-1.322219304i$$

$$x=\frac{-1+\sqrt{1007}i}{24}\approx -0.041666667+1.322219304i$$

Steps Using the Quadratic Formula

Combine -2x^{2} and -4x^{2} to get -6x^{2}.

$$-6x^{2}+9x-9+2x+6-6x^{2}-12x-18=0$$

Combine 9x and 2x to get 11x.

$$-6x^{2}+11x-9+6-6x^{2}-12x-18=0$$

Add -9 and 6 to get -3.

$$-6x^{2}+11x-3-6x^{2}-12x-18=0$$

Combine -6x^{2} and -6x^{2} to get -12x^{2}.

$$-12x^{2}+11x-3-12x-18=0$$

Combine 11x and -12x to get -x.

$$-12x^{2}-x-3-18=0$$

Subtract 18 from -3 to get -21.

$$-12x^{2}-x-21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -12 for a, -1 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-12\right)\left(-21\right)}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$x=\frac{-\left(-1\right)±\sqrt{1+48\left(-21\right)}}{2\left(-12\right)}$$

Multiply 48 times -21.

$$x=\frac{-\left(-1\right)±\sqrt{1-1008}}{2\left(-12\right)}$$

Add 1 to -1008.

$$x=\frac{-\left(-1\right)±\sqrt{-1007}}{2\left(-12\right)}$$

Take the square root of -1007.

$$x=\frac{-\left(-1\right)±\sqrt{1007}i}{2\left(-12\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{1007}i}{2\left(-12\right)}$$

Multiply 2 times -12.

$$x=\frac{1±\sqrt{1007}i}{-24}$$

Now solve the equation x=\frac{1±\sqrt{1007}i}{-24} when ± is plus. Add 1 to i\sqrt{1007}.

$$x=\frac{1+\sqrt{1007}i}{-24}$$

Divide 1+i\sqrt{1007} by -24.

$$x=\frac{-\sqrt{1007}i-1}{24}$$

Now solve the equation x=\frac{1±\sqrt{1007}i}{-24} when ± is minus. Subtract i\sqrt{1007} from 1.

$$x=\frac{-\sqrt{1007}i+1}{-24}$$

Divide 1-i\sqrt{1007} by -24.

$$x=\frac{-1+\sqrt{1007}i}{24}$$

The equation is now solved.

$$x=\frac{-\sqrt{1007}i-1}{24} x=\frac{-1+\sqrt{1007}i}{24}$$

Steps for Completing the Square

Combine -2x^{2} and -4x^{2} to get -6x^{2}.

$$-6x^{2}+9x-9+2x+6-6x^{2}-12x-18=0$$

Combine 9x and 2x to get 11x.

$$-6x^{2}+11x-9+6-6x^{2}-12x-18=0$$

Add -9 and 6 to get -3.

$$-6x^{2}+11x-3-6x^{2}-12x-18=0$$

Combine -6x^{2} and -6x^{2} to get -12x^{2}.

$$-12x^{2}+11x-3-12x-18=0$$

Combine 11x and -12x to get -x.

$$-12x^{2}-x-3-18=0$$

Subtract 18 from -3 to get -21.

$$-12x^{2}-x-21=0$$

Add 21 to both sides. Anything plus zero gives itself.

$$-12x^{2}-x=21$$

Divide both sides by -12.

$$\frac{-12x^{2}-x}{-12}=\frac{21}{-12}$$

Dividing by -12 undoes the multiplication by -12.

$$x^{2}+\left(-\frac{1}{-12}\right)x=\frac{21}{-12}$$

Divide -1 by -12.

$$x^{2}+\frac{1}{12}x=\frac{21}{-12}$$

Reduce the fraction \frac{21}{-12} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{1}{12}x=-\frac{7}{4}$$

Divide \frac{1}{12}, the coefficient of the x term, by 2 to get \frac{1}{24}. Then add the square of \frac{1}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{12}x+\left(\frac{1}{24}\right)^{2}=-\frac{7}{4}+\left(\frac{1}{24}\right)^{2}$$

Square \frac{1}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{12}x+\frac{1}{576}=-\frac{7}{4}+\frac{1}{576}$$

Add -\frac{7}{4} to \frac{1}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{12}x+\frac{1}{576}=-\frac{1007}{576}$$

Factor x^{2}+\frac{1}{12}x+\frac{1}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{24}\right)^{2}=-\frac{1007}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{24}\right)^{2}}=\sqrt{-\frac{1007}{576}}$$

Simplify.

$$x+\frac{1}{24}=\frac{\sqrt{1007}i}{24} x+\frac{1}{24}=-\frac{\sqrt{1007}i}{24}$$

Subtract \frac{1}{24} from both sides of the equation.

$$x=\frac{-1+\sqrt{1007}i}{24} x=\frac{-\sqrt{1007}i-1}{24}$$