Solve 2x^2-2x-x+1-4x^2-9=6-5x

$$2 x ^ { 2 } - 2 x - x + 1 - 4 x ^ { 2 } - 9 = 6 - 5 x$$

Solve for x (complex solution)

$$x=\frac{1+3\sqrt{3}i}{2}\approx 0.5+2.598076211i$$

$$x=\frac{-3\sqrt{3}i+1}{2}\approx 0.5-2.598076211i$$

Steps Using the Quadratic Formula

Combine -2x and -x to get -3x.

$$2x^{2}-3x+1-4x^{2}-9=6-5x$$

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

$$-2x^{2}-3x+1-9=6-5x$$

Subtract 9 from 1 to get -8.

$$-2x^{2}-3x-8=6-5x$$

Subtract 6 from both sides.

$$-2x^{2}-3x-8-6=-5x$$

Subtract 6 from -8 to get -14.

$$-2x^{2}-3x-14=-5x$$

Add 5x to both sides.

$$-2x^{2}-3x-14+5x=0$$

Combine -3x and 5x to get 2x.

$$-2x^{2}+2x-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 2 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-2\right)\left(-14\right)}}{2\left(-2\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-2\right)\left(-14\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-2±\sqrt{4+8\left(-14\right)}}{2\left(-2\right)}$$

Multiply 8 times -14.

$$x=\frac{-2±\sqrt{4-112}}{2\left(-2\right)}$$

Add 4 to -112.

$$x=\frac{-2±\sqrt{-108}}{2\left(-2\right)}$$

Take the square root of -108.

$$x=\frac{-2±6\sqrt{3}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-2±6\sqrt{3}i}{-4}$$

Now solve the equation x=\frac{-2±6\sqrt{3}i}{-4} when ± is plus. Add -2 to 6i\sqrt{3}.

$$x=\frac{-2+6\sqrt{3}i}{-4}$$

Divide -2+6i\sqrt{3} by -4.

$$x=\frac{-3\sqrt{3}i+1}{2}$$

Now solve the equation x=\frac{-2±6\sqrt{3}i}{-4} when ± is minus. Subtract 6i\sqrt{3} from -2.

$$x=\frac{-6\sqrt{3}i-2}{-4}$$

Divide -2-6i\sqrt{3} by -4.

$$x=\frac{1+3\sqrt{3}i}{2}$$

The equation is now solved.

$$x=\frac{-3\sqrt{3}i+1}{2} x=\frac{1+3\sqrt{3}i}{2}$$

Steps for Completing the Square

Combine -2x and -x to get -3x.

$$2x^{2}-3x+1-4x^{2}-9=6-5x$$

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

$$-2x^{2}-3x+1-9=6-5x$$

Subtract 9 from 1 to get -8.

$$-2x^{2}-3x-8=6-5x$$

Add 5x to both sides.

$$-2x^{2}-3x-8+5x=6$$

Combine -3x and 5x to get 2x.

$$-2x^{2}+2x-8=6$$

Add 8 to both sides.

$$-2x^{2}+2x=6+8$$

Add 6 and 8 to get 14.

$$-2x^{2}+2x=14$$

Divide both sides by -2.

$$\frac{-2x^{2}+2x}{-2}=\frac{14}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{2}{-2}x=\frac{14}{-2}$$

Divide 2 by -2.

$$x^{2}-x=\frac{14}{-2}$$

Divide 14 by -2.

$$x^{2}-x=-7$$

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.

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

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

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

Add -7 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=-\frac{27}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=-\frac{27}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{27}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{3\sqrt{3}i}{2} x-\frac{1}{2}=-\frac{3\sqrt{3}i}{2}$$

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

$$x=\frac{1+3\sqrt{3}i}{2} x=\frac{-3\sqrt{3}i+1}{2}$$