Solve 3^3-6x^2+12x-8=0

$${ 3 }^{ 3 } -6 { x }^{ 2 } +12x-8=0$$

Solve for x

$$x=\frac{5\sqrt{6}}{6}+1\approx 3.041241452$$

$$x=-\frac{5\sqrt{6}}{6}+1\approx -1.041241452$$

Steps Using the Quadratic Formula

Calculate 3 to the power of 3 and get 27.

$$27-6x^{2}+12x-8=0$$

Subtract 8 from 27 to get 19.

$$19-6x^{2}+12x=0$$

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.

$$-6x^{2}+12x+19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 12 for b, and 19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-6\right)\times 19}}{2\left(-6\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-6\right)\times 19}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-12±\sqrt{144+24\times 19}}{2\left(-6\right)}$$

Multiply 24 times 19.

$$x=\frac{-12±\sqrt{144+456}}{2\left(-6\right)}$$

Add 144 to 456.

$$x=\frac{-12±\sqrt{600}}{2\left(-6\right)}$$

Take the square root of 600.

$$x=\frac{-12±10\sqrt{6}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-12±10\sqrt{6}}{-12}$$

Now solve the equation x=\frac{-12±10\sqrt{6}}{-12} when ± is plus. Add -12 to 10\sqrt{6}.

$$x=\frac{10\sqrt{6}-12}{-12}$$

Divide -12+10\sqrt{6} by -12.

$$x=-\frac{5\sqrt{6}}{6}+1$$

Now solve the equation x=\frac{-12±10\sqrt{6}}{-12} when ± is minus. Subtract 10\sqrt{6} from -12.

$$x=\frac{-10\sqrt{6}-12}{-12}$$

Divide -12-10\sqrt{6} by -12.

$$x=\frac{5\sqrt{6}}{6}+1$$

The equation is now solved.

$$x=-\frac{5\sqrt{6}}{6}+1 x=\frac{5\sqrt{6}}{6}+1$$

Steps for Completing the Square

Calculate 3 to the power of 3 and get 27.

$$27-6x^{2}+12x-8=0$$

Subtract 8 from 27 to get 19.

$$19-6x^{2}+12x=0$$

Subtract 19 from both sides. Anything subtracted from zero gives its negation.

$$-6x^{2}+12x=-19$$

Divide both sides by -6.

$$\frac{-6x^{2}+12x}{-6}=-\frac{19}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{12}{-6}x=-\frac{19}{-6}$$

Divide 12 by -6.

$$x^{2}-2x=-\frac{19}{-6}$$

Divide -19 by -6.

$$x^{2}-2x=\frac{19}{6}$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=\frac{19}{6}+1$$

Add \frac{19}{6} to 1.

$$x^{2}-2x+1=\frac{25}{6}$$

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

$$\left(x-1\right)^{2}=\frac{25}{6}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{25}{6}}$$

Simplify.

$$x-1=\frac{5\sqrt{6}}{6} x-1=-\frac{5\sqrt{6}}{6}$$

Add 1 to both sides of the equation.

$$x=\frac{5\sqrt{6}}{6}+1 x=-\frac{5\sqrt{6}}{6}+1$$