Solve 6x^2+39=5+12x

$$6 x ^ { 2 } + 39 = 5 + 12 x$$

Solve for x (complex solution)

$$x=\frac{\sqrt{42}i}{3}+1\approx 1+2.160246899i$$

$$x=-\frac{\sqrt{42}i}{3}+1\approx 1-2.160246899i$$

Steps Using the Quadratic Formula

Subtract 5 from both sides.

$$6x^{2}+39-5=12x$$

Subtract 5 from 39 to get 34.

$$6x^{2}+34=12x$$

Subtract 12x from both sides.

$$6x^{2}+34-12x=0$$

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.

$$6x^{2}-12x+34=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -12 for b, and 34 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 6\times 34}}{2\times 6}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 6\times 34}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-12\right)±\sqrt{144-24\times 34}}{2\times 6}$$

Multiply -24 times 34.

$$x=\frac{-\left(-12\right)±\sqrt{144-816}}{2\times 6}$$

Add 144 to -816.

$$x=\frac{-\left(-12\right)±\sqrt{-672}}{2\times 6}$$

Take the square root of -672.

$$x=\frac{-\left(-12\right)±4\sqrt{42}i}{2\times 6}$$

The opposite of -12 is 12.

$$x=\frac{12±4\sqrt{42}i}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{12±4\sqrt{42}i}{12}$$

Now solve the equation x=\frac{12±4\sqrt{42}i}{12} when ± is plus. Add 12 to 4i\sqrt{42}.

$$x=\frac{12+4\sqrt{42}i}{12}$$

Divide 12+4i\sqrt{42} by 12.

$$x=\frac{\sqrt{42}i}{3}+1$$

Now solve the equation x=\frac{12±4\sqrt{42}i}{12} when ± is minus. Subtract 4i\sqrt{42} from 12.

$$x=\frac{-4\sqrt{42}i+12}{12}$$

Divide 12-4i\sqrt{42} by 12.

$$x=-\frac{\sqrt{42}i}{3}+1$$

The equation is now solved.

$$x=\frac{\sqrt{42}i}{3}+1 x=-\frac{\sqrt{42}i}{3}+1$$

Steps for Completing the Square

Subtract 12x from both sides.

$$6x^{2}+39-12x=5$$

Subtract 39 from both sides.

$$6x^{2}-12x=5-39$$

Subtract 39 from 5 to get -34.

$$6x^{2}-12x=-34$$

Divide both sides by 6.

$$\frac{6x^{2}-12x}{6}=-\frac{34}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{12}{6}\right)x=-\frac{34}{6}$$

Divide -12 by 6.

$$x^{2}-2x=-\frac{34}{6}$$

Reduce the fraction \frac{-34}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}-2x=-\frac{17}{3}$$

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.

$$x^{2}-2x+1=-\frac{17}{3}+1$$

Add -\frac{17}{3} to 1.

$$x^{2}-2x+1=-\frac{14}{3}$$

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}.

$$\left(x-1\right)^{2}=-\frac{14}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-\frac{14}{3}}$$

Simplify.

$$x-1=\frac{\sqrt{42}i}{3} x-1=-\frac{\sqrt{42}i}{3}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{42}i}{3}+1 x=-\frac{\sqrt{42}i}{3}+1$$