Solve x=-2(3x^2-5x+4)

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

Solve for x (complex solution)

$$x=\frac{\sqrt{111}i}{12}+\frac{3}{4}\approx 0.75+0.877971146i$$

$$x=-\frac{\sqrt{111}i}{12}+\frac{3}{4}\approx 0.75-0.877971146i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -2 by 3x^{2}-5x+4.

$$x=-6x^{2}+10x-8$$

Add 6x^{2} to both sides.

$$x+6x^{2}=10x-8$$

Subtract 10x from both sides.

$$x+6x^{2}-10x=-8$$

Combine x and -10x to get -9x.

$$-9x+6x^{2}=-8$$

Add 8 to both sides.

$$-9x+6x^{2}+8=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$6x^{2}-9x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -9 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 6\times 8}}{2\times 6}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 6\times 8}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-9\right)±\sqrt{81-24\times 8}}{2\times 6}$$

Multiply -24 times 8.

$$x=\frac{-\left(-9\right)±\sqrt{81-192}}{2\times 6}$$

Add 81 to -192.

$$x=\frac{-\left(-9\right)±\sqrt{-111}}{2\times 6}$$

Take the square root of -111.

$$x=\frac{-\left(-9\right)±\sqrt{111}i}{2\times 6}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{111}i}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{9±\sqrt{111}i}{12}$$

Now solve the equation x=\frac{9±\sqrt{111}i}{12} when ± is plus. Add 9 to i\sqrt{111}.

$$x=\frac{9+\sqrt{111}i}{12}$$

Divide 9+i\sqrt{111} by 12.

$$x=\frac{\sqrt{111}i}{12}+\frac{3}{4}$$

Now solve the equation x=\frac{9±\sqrt{111}i}{12} when ± is minus. Subtract i\sqrt{111} from 9.

$$x=\frac{-\sqrt{111}i+9}{12}$$

Divide 9-i\sqrt{111} by 12.

$$x=-\frac{\sqrt{111}i}{12}+\frac{3}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{111}i}{12}+\frac{3}{4} x=-\frac{\sqrt{111}i}{12}+\frac{3}{4}$$

Steps for Completing the Square

Use the distributive property to multiply -2 by 3x^{2}-5x+4.

$$x=-6x^{2}+10x-8$$

Add 6x^{2} to both sides.

$$x+6x^{2}=10x-8$$

Subtract 10x from both sides.

$$x+6x^{2}-10x=-8$$

Combine x and -10x to get -9x.

$$-9x+6x^{2}=-8$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$6x^{2}-9x=-8$$

Divide both sides by 6.

$$\frac{6x^{2}-9x}{6}=-\frac{8}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{9}{6}\right)x=-\frac{8}{6}$$

Reduce the fraction \frac{-9}{6} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{3}{2}x=-\frac{8}{6}$$

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

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

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

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

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{4}{3}+\frac{9}{16}$$

Add -\frac{4}{3} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{37}{48}$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{4}\right)^{2}=-\frac{37}{48}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{-\frac{37}{48}}$$

Simplify.

$$x-\frac{3}{4}=\frac{\sqrt{111}i}{12} x-\frac{3}{4}=-\frac{\sqrt{111}i}{12}$$

Add \frac{3}{4} to both sides of the equation.

$$x=\frac{\sqrt{111}i}{12}+\frac{3}{4} x=-\frac{\sqrt{111}i}{12}+\frac{3}{4}$$