Solve 2x^2-4x=5x^2-3x+1

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

Solve for x (complex solution)

$$x=\frac{-\sqrt{11}i-1}{6}\approx -0.166666667-0.552770798i$$

$$x=\frac{-1+\sqrt{11}i}{6}\approx -0.166666667+0.552770798i$$

Steps Using the Quadratic Formula

Subtract 5x^{2} from both sides.

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

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

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

Add 3x to both sides.

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

Combine -4x and 3x to get -x.

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

Subtract 1 from both sides.

$$-3x^{2}-x-1=0$$

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

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

Multiply -4 times -3.

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

Multiply 12 times -1.

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

Add 1 to -12.

$$x=\frac{-\left(-1\right)±\sqrt{-11}}{2\left(-3\right)}$$

Take the square root of -11.

$$x=\frac{-\left(-1\right)±\sqrt{11}i}{2\left(-3\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{11}i}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{1±\sqrt{11}i}{-6}$$

Now solve the equation x=\frac{1±\sqrt{11}i}{-6} when ± is plus. Add 1 to i\sqrt{11}.

$$x=\frac{1+\sqrt{11}i}{-6}$$

Divide 1+i\sqrt{11} by -6.

$$x=\frac{-\sqrt{11}i-1}{6}$$

Now solve the equation x=\frac{1±\sqrt{11}i}{-6} when ± is minus. Subtract i\sqrt{11} from 1.

$$x=\frac{-\sqrt{11}i+1}{-6}$$

Divide 1-i\sqrt{11} by -6.

$$x=\frac{-1+\sqrt{11}i}{6}$$

The equation is now solved.

$$x=\frac{-\sqrt{11}i-1}{6} x=\frac{-1+\sqrt{11}i}{6}$$

Steps for Completing the Square

Subtract 5x^{2} from both sides.

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

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

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

Add 3x to both sides.

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

Combine -4x and 3x to get -x.

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

Divide both sides by -3.

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

Dividing by -3 undoes the multiplication by -3.

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

Divide -1 by -3.

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

Divide 1 by -3.

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

Divide \frac{1}{3}, the coefficient of the x term, by 2 to get \frac{1}{6}. Then add the square of \frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{3}x+\left(\frac{1}{6}\right)^{2}=-\frac{1}{3}+\left(\frac{1}{6}\right)^{2}$$

Square \frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=-\frac{1}{3}+\frac{1}{36}$$

Add -\frac{1}{3} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=-\frac{11}{36}$$

Factor x^{2}+\frac{1}{3}x+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{6}\right)^{2}=-\frac{11}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{6}\right)^{2}}=\sqrt{-\frac{11}{36}}$$

Simplify.

$$x+\frac{1}{6}=\frac{\sqrt{11}i}{6} x+\frac{1}{6}=-\frac{\sqrt{11}i}{6}$$

Subtract \frac{1}{6} from both sides of the equation.

$$x=\frac{-1+\sqrt{11}i}{6} x=\frac{-\sqrt{11}i-1}{6}$$