Solve x^2-x+4=2x^2-4x+7

$$x ^ { 2 } - x + 4 = 2 x ^ { 2 } - 4 x + 7$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{3}i+3}{2}\approx 1.5-0.866025404i$$

$$x=\frac{3+\sqrt{3}i}{2}\approx 1.5+0.866025404i$$

Steps Using the Quadratic Formula

Subtract 2x^{2} from both sides.

$$x^{2}-x+4-2x^{2}=-4x+7$$

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

$$-x^{2}-x+4=-4x+7$$

Add 4x to both sides.

$$-x^{2}-x+4+4x=7$$

Combine -x and 4x to get 3x.

$$-x^{2}+3x+4=7$$

Subtract 7 from both sides.

$$-x^{2}+3x+4-7=0$$

Subtract 7 from 4 to get -3.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 3 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-3±\sqrt{9+4\left(-3\right)}}{2\left(-1\right)}$$

Multiply 4 times -3.

$$x=\frac{-3±\sqrt{9-12}}{2\left(-1\right)}$$

Add 9 to -12.

$$x=\frac{-3±\sqrt{-3}}{2\left(-1\right)}$$

Take the square root of -3.

$$x=\frac{-3±\sqrt{3}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-3±\sqrt{3}i}{-2}$$

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

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

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

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

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

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

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

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

The equation is now solved.

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

Steps for Completing the Square

Subtract 2x^{2} from both sides.

$$x^{2}-x+4-2x^{2}=-4x+7$$

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

$$-x^{2}-x+4=-4x+7$$

Add 4x to both sides.

$$-x^{2}-x+4+4x=7$$

Combine -x and 4x to get 3x.

$$-x^{2}+3x+4=7$$

Subtract 4 from both sides.

$$-x^{2}+3x=7-4$$

Subtract 4 from 7 to get 3.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 3 by -1.

$$x^{2}-3x=\frac{3}{-1}$$

Divide 3 by -1.

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

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

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

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

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

Add -3 to \frac{9}{4}.

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

Factor x^{2}-3x+\frac{9}{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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add \frac{3}{2} to both sides of the equation.

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