Solve (2x-1)(-3x+4)=-6x+11x+4

$$( 2 x - 1 ) ( - 3 x + 4 ) = - 6 x + 11 x + 4$$

Solve for x (complex solution)

$$x=\frac{\sqrt{39}i}{6}+\frac{1}{2}\approx 0.5+1.040833i$$

$$x=-\frac{\sqrt{39}i}{6}+\frac{1}{2}\approx 0.5-1.040833i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x-1 by -3x+4 and combine like terms.

$$-6x^{2}+11x-4=-6x+11x+4$$

Combine -6x and 11x to get 5x.

$$-6x^{2}+11x-4=5x+4$$

Subtract 5x from both sides.

$$-6x^{2}+11x-4-5x=4$$

Combine 11x and -5x to get 6x.

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

Subtract 4 from both sides.

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

Subtract 4 from -4 to get -8.

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

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

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

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-6\right)\left(-8\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-6±\sqrt{36+24\left(-8\right)}}{2\left(-6\right)}$$

Multiply 24 times -8.

$$x=\frac{-6±\sqrt{36-192}}{2\left(-6\right)}$$

Add 36 to -192.

$$x=\frac{-6±\sqrt{-156}}{2\left(-6\right)}$$

Take the square root of -156.

$$x=\frac{-6±2\sqrt{39}i}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-6±2\sqrt{39}i}{-12}$$

Now solve the equation x=\frac{-6±2\sqrt{39}i}{-12} when ± is plus. Add -6 to 2i\sqrt{39}.

$$x=\frac{-6+2\sqrt{39}i}{-12}$$

Divide -6+2i\sqrt{39} by -12.

$$x=-\frac{\sqrt{39}i}{6}+\frac{1}{2}$$

Now solve the equation x=\frac{-6±2\sqrt{39}i}{-12} when ± is minus. Subtract 2i\sqrt{39} from -6.

$$x=\frac{-2\sqrt{39}i-6}{-12}$$

Divide -6-2i\sqrt{39} by -12.

$$x=\frac{\sqrt{39}i}{6}+\frac{1}{2}$$

The equation is now solved.

$$x=-\frac{\sqrt{39}i}{6}+\frac{1}{2} x=\frac{\sqrt{39}i}{6}+\frac{1}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 2x-1 by -3x+4 and combine like terms.

$$-6x^{2}+11x-4=-6x+11x+4$$

Combine -6x and 11x to get 5x.

$$-6x^{2}+11x-4=5x+4$$

Subtract 5x from both sides.

$$-6x^{2}+11x-4-5x=4$$

Combine 11x and -5x to get 6x.

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

Add 4 to both sides.

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

Add 4 and 4 to get 8.

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

Divide both sides by -6.

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

Dividing by -6 undoes the multiplication by -6.

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

Divide 6 by -6.

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

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

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

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.

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

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

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

Add -\frac{4}{3} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=-\frac{13}{12}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=-\frac{13}{12}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{13}{12}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{39}i}{6} x-\frac{1}{2}=-\frac{\sqrt{39}i}{6}$$

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

$$x=\frac{\sqrt{39}i}{6}+\frac{1}{2} x=-\frac{\sqrt{39}i}{6}+\frac{1}{2}$$