Solve -3x^2-3x+11=2x

$$- 3 x ^ { 2 } - 3 x + 11 = 2 x$$

Solve for x

$$x = \frac{\sqrt{157} - 5}{6} \approx 1.254994014$$

$$x=\frac{-\sqrt{157}-5}{6}\approx -2.921660681$$

Steps Using the Quadratic Formula

Subtract 2x from both sides.

$$-3x^{2}-3x+11-2x=0$$

Combine -3x and -2x to get -5x.

$$-3x^{2}-5x+11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -5 for b, and 11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-3\right)\times 11}}{2\left(-3\right)}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-3\right)\times 11}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-5\right)±\sqrt{25+12\times 11}}{2\left(-3\right)}$$

Multiply 12 times 11.

$$x=\frac{-\left(-5\right)±\sqrt{25+132}}{2\left(-3\right)}$$

Add 25 to 132.

$$x=\frac{-\left(-5\right)±\sqrt{157}}{2\left(-3\right)}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{157}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{5±\sqrt{157}}{-6}$$

Now solve the equation x=\frac{5±\sqrt{157}}{-6} when ± is plus. Add 5 to \sqrt{157}.

$$x=\frac{\sqrt{157}+5}{-6}$$

Divide 5+\sqrt{157} by -6.

$$x=\frac{-\sqrt{157}-5}{6}$$

Now solve the equation x=\frac{5±\sqrt{157}}{-6} when ± is minus. Subtract \sqrt{157} from 5.

$$x=\frac{5-\sqrt{157}}{-6}$$

Divide 5-\sqrt{157} by -6.

$$x=\frac{\sqrt{157}-5}{6}$$

The equation is now solved.

$$x=\frac{-\sqrt{157}-5}{6} x=\frac{\sqrt{157}-5}{6}$$

Steps for Completing the Square

Subtract 2x from both sides.

$$-3x^{2}-3x+11-2x=0$$

Combine -3x and -2x to get -5x.

$$-3x^{2}-5x+11=0$$

Subtract 11 from both sides. Anything subtracted from zero gives its negation.

$$-3x^{2}-5x=-11$$

Divide both sides by -3.

$$\frac{-3x^{2}-5x}{-3}=-\frac{11}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{5}{-3}\right)x=-\frac{11}{-3}$$

Divide -5 by -3.

$$x^{2}+\frac{5}{3}x=-\frac{11}{-3}$$

Divide -11 by -3.

$$x^{2}+\frac{5}{3}x=\frac{11}{3}$$

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.

$$x^{2}+\frac{5}{3}x+\left(\frac{5}{6}\right)^{2}=\frac{11}{3}+\left(\frac{5}{6}\right)^{2}$$

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

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{11}{3}+\frac{25}{36}$$

Add \frac{11}{3} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{157}{36}$$

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}.

$$\left(x+\frac{5}{6}\right)^{2}=\frac{157}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{6}\right)^{2}}=\sqrt{\frac{157}{36}}$$

Simplify.

$$x+\frac{5}{6}=\frac{\sqrt{157}}{6} x+\frac{5}{6}=-\frac{\sqrt{157}}{6}$$

Subtract \frac{5}{6} from both sides of the equation.

$$x=\frac{\sqrt{157}-5}{6} x=\frac{-\sqrt{157}-5}{6}$$