Solve x^2-4(x^2+x+2)=3x^2+4x+4

$$x ^ { 2 } - 4 ( x ^ { 2 } + x + 2 ) = 3 x ^ { 2 } + 4 x + 4$$

Solve for x (complex solution)

$$x=\frac{-2+\sqrt{14}i}{3}\approx -0.666666667+1.247219129i$$

$$x=\frac{-\sqrt{14}i-2}{3}\approx -0.666666667-1.247219129i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -4 by x^{2}+x+2.

$$x^{2}-4x^{2}-4x-8=3x^{2}+4x+4$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-4x-8=3x^{2}+4x+4$$

Subtract 3x^{2} from both sides.

$$-3x^{2}-4x-8-3x^{2}=4x+4$$

Combine -3x^{2} and -3x^{2} to get -6x^{2}.

$$-6x^{2}-4x-8=4x+4$$

Subtract 4x from both sides.

$$-6x^{2}-4x-8-4x=4$$

Combine -4x and -4x to get -8x.

$$-6x^{2}-8x-8=4$$

Subtract 4 from both sides.

$$-6x^{2}-8x-8-4=0$$

Subtract 4 from -8 to get -12.

$$-6x^{2}-8x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, -8 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-6\right)\left(-12\right)}}{2\left(-6\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-6\right)\left(-12\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-\left(-8\right)±\sqrt{64+24\left(-12\right)}}{2\left(-6\right)}$$

Multiply 24 times -12.

$$x=\frac{-\left(-8\right)±\sqrt{64-288}}{2\left(-6\right)}$$

Add 64 to -288.

$$x=\frac{-\left(-8\right)±\sqrt{-224}}{2\left(-6\right)}$$

Take the square root of -224.

$$x=\frac{-\left(-8\right)±4\sqrt{14}i}{2\left(-6\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±4\sqrt{14}i}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{8±4\sqrt{14}i}{-12}$$

Now solve the equation x=\frac{8±4\sqrt{14}i}{-12} when ± is plus. Add 8 to 4i\sqrt{14}.

$$x=\frac{8+4\sqrt{14}i}{-12}$$

Divide 8+4i\sqrt{14} by -12.

$$x=\frac{-\sqrt{14}i-2}{3}$$

Now solve the equation x=\frac{8±4\sqrt{14}i}{-12} when ± is minus. Subtract 4i\sqrt{14} from 8.

$$x=\frac{-4\sqrt{14}i+8}{-12}$$

Divide 8-4i\sqrt{14} by -12.

$$x=\frac{-2+\sqrt{14}i}{3}$$

The equation is now solved.

$$x=\frac{-\sqrt{14}i-2}{3} x=\frac{-2+\sqrt{14}i}{3}$$

Steps for Completing the Square

Use the distributive property to multiply -4 by x^{2}+x+2.

$$x^{2}-4x^{2}-4x-8=3x^{2}+4x+4$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-4x-8=3x^{2}+4x+4$$

Subtract 3x^{2} from both sides.

$$-3x^{2}-4x-8-3x^{2}=4x+4$$

Combine -3x^{2} and -3x^{2} to get -6x^{2}.

$$-6x^{2}-4x-8=4x+4$$

Subtract 4x from both sides.

$$-6x^{2}-4x-8-4x=4$$

Combine -4x and -4x to get -8x.

$$-6x^{2}-8x-8=4$$

Add 8 to both sides.

$$-6x^{2}-8x=4+8$$

Add 4 and 8 to get 12.

$$-6x^{2}-8x=12$$

Divide both sides by -6.

$$\frac{-6x^{2}-8x}{-6}=\frac{12}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\left(-\frac{8}{-6}\right)x=\frac{12}{-6}$$

Reduce the fraction \frac{-8}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{4}{3}x=\frac{12}{-6}$$

Divide 12 by -6.

$$x^{2}+\frac{4}{3}x=-2$$

Divide \frac{4}{3}, the coefficient of the x term, by 2 to get \frac{2}{3}. Then add the square of \frac{2}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{4}{3}x+\left(\frac{2}{3}\right)^{2}=-2+\left(\frac{2}{3}\right)^{2}$$

Square \frac{2}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=-2+\frac{4}{9}$$

Add -2 to \frac{4}{9}.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=-\frac{14}{9}$$

Factor x^{2}+\frac{4}{3}x+\frac{4}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2}{3}\right)^{2}=-\frac{14}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2}{3}\right)^{2}}=\sqrt{-\frac{14}{9}}$$

Simplify.

$$x+\frac{2}{3}=\frac{\sqrt{14}i}{3} x+\frac{2}{3}=-\frac{\sqrt{14}i}{3}$$

Subtract \frac{2}{3} from both sides of the equation.

$$x=\frac{-2+\sqrt{14}i}{3} x=\frac{-\sqrt{14}i-2}{3}$$