Solve 3x^2+6=5x

$$3 x ^ { 2 } + 6 = 5 x$$

Solve for x (complex solution)

$$x=\frac{5+\sqrt{47}i}{6}\approx 0.833333333+1.1426091i$$

$$x=\frac{-\sqrt{47}i+5}{6}\approx 0.833333333-1.1426091i$$

Steps Using the Quadratic Formula

Subtract 5x from both sides.

$$3x^{2}+6-5x=0$$

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.

$$3x^{2}-5x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -5 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 3\times 6}}{2\times 3}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 3\times 6}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-5\right)±\sqrt{25-12\times 6}}{2\times 3}$$

Multiply -12 times 6.

$$x=\frac{-\left(-5\right)±\sqrt{25-72}}{2\times 3}$$

Add 25 to -72.

$$x=\frac{-\left(-5\right)±\sqrt{-47}}{2\times 3}$$

Take the square root of -47.

$$x=\frac{-\left(-5\right)±\sqrt{47}i}{2\times 3}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{47}i}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{5±\sqrt{47}i}{6}$$

Now solve the equation x=\frac{5±\sqrt{47}i}{6} when ± is plus. Add 5 to i\sqrt{47}.

$$x=\frac{5+\sqrt{47}i}{6}$$

Now solve the equation x=\frac{5±\sqrt{47}i}{6} when ± is minus. Subtract i\sqrt{47} from 5.

$$x=\frac{-\sqrt{47}i+5}{6}$$

The equation is now solved.

$$x=\frac{5+\sqrt{47}i}{6} x=\frac{-\sqrt{47}i+5}{6}$$

Steps for Completing the Square

Subtract 5x from both sides.

$$3x^{2}+6-5x=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$3x^{2}-5x=-6$$

Divide both sides by 3.

$$\frac{3x^{2}-5x}{3}=-\frac{6}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{5}{3}x=-\frac{6}{3}$$

Divide -6 by 3.

$$x^{2}-\frac{5}{3}x=-2$$

Divide -\frac{5}{3}, the coefficient of the x term, by 2 to get -\frac{5}{6}. Then add the square of -\frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{3}x+\left(-\frac{5}{6}\right)^{2}=-2+\left(-\frac{5}{6}\right)^{2}$$

Square -\frac{5}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=-2+\frac{25}{36}$$

Add -2 to \frac{25}{36}.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=-\frac{47}{36}$$

Factor x^{2}-\frac{5}{3}x+\frac{25}{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}.

$$\left(x-\frac{5}{6}\right)^{2}=-\frac{47}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{6}\right)^{2}}=\sqrt{-\frac{47}{36}}$$

Simplify.

$$x-\frac{5}{6}=\frac{\sqrt{47}i}{6} x-\frac{5}{6}=-\frac{\sqrt{47}i}{6}$$

Add \frac{5}{6} to both sides of the equation.

$$x=\frac{5+\sqrt{47}i}{6} x=\frac{-\sqrt{47}i+5}{6}$$