Solve (11)2x^2+5/3x-2=0

$$( 11 ) 2 x ^ { 2 } + \frac { 5 } { 3 } x - 2 = 0$$

Solve for x

$$x=\frac{\sqrt{1609}-5}{132}\approx 0.266002593$$

$$x=\frac{-\sqrt{1609}-5}{132}\approx -0.341760168$$

Steps Using the Quadratic Formula

Multiply 11 and 2 to get 22.

$$22x^{2}+\frac{5}{3}x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 22 for a, \frac{5}{3} for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{5}{3}±\sqrt{\left(\frac{5}{3}\right)^{2}-4\times 22\left(-2\right)}}{2\times 22}$$

Square \frac{5}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{5}{3}±\sqrt{\frac{25}{9}-4\times 22\left(-2\right)}}{2\times 22}$$

Multiply -4 times 22.

$$x=\frac{-\frac{5}{3}±\sqrt{\frac{25}{9}-88\left(-2\right)}}{2\times 22}$$

Multiply -88 times -2.

$$x=\frac{-\frac{5}{3}±\sqrt{\frac{25}{9}+176}}{2\times 22}$$

Add \frac{25}{9} to 176.

$$x=\frac{-\frac{5}{3}±\sqrt{\frac{1609}{9}}}{2\times 22}$$

Take the square root of \frac{1609}{9}.

$$x=\frac{-\frac{5}{3}±\frac{\sqrt{1609}}{3}}{2\times 22}$$

Multiply 2 times 22.

$$x=\frac{-\frac{5}{3}±\frac{\sqrt{1609}}{3}}{44}$$

Now solve the equation x=\frac{-\frac{5}{3}±\frac{\sqrt{1609}}{3}}{44} when ± is plus. Add -\frac{5}{3} to \frac{\sqrt{1609}}{3}.

$$x=\frac{\sqrt{1609}-5}{3\times 44}$$

Divide \frac{-5+\sqrt{1609}}{3} by 44.

$$x=\frac{\sqrt{1609}-5}{132}$$

Now solve the equation x=\frac{-\frac{5}{3}±\frac{\sqrt{1609}}{3}}{44} when ± is minus. Subtract \frac{\sqrt{1609}}{3} from -\frac{5}{3}.

$$x=\frac{-\sqrt{1609}-5}{3\times 44}$$

Divide \frac{-5-\sqrt{1609}}{3} by 44.

$$x=\frac{-\sqrt{1609}-5}{132}$$

The equation is now solved.

$$x=\frac{\sqrt{1609}-5}{132} x=\frac{-\sqrt{1609}-5}{132}$$

Steps for Completing the Square

Multiply 11 and 2 to get 22.

$$22x^{2}+\frac{5}{3}x-2=0$$

Add 2 to both sides. Anything plus zero gives itself.

$$22x^{2}+\frac{5}{3}x=2$$

Divide both sides by 22.

$$\frac{22x^{2}+\frac{5}{3}x}{22}=\frac{2}{22}$$

Dividing by 22 undoes the multiplication by 22.

$$x^{2}+\frac{\frac{5}{3}}{22}x=\frac{2}{22}$$

Divide \frac{5}{3} by 22.

$$x^{2}+\frac{5}{66}x=\frac{2}{22}$$

Reduce the fraction \frac{2}{22} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{5}{66}x=\frac{1}{11}$$

Divide \frac{5}{66}, the coefficient of the x term, by 2 to get \frac{5}{132}. Then add the square of \frac{5}{132} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{66}x+\left(\frac{5}{132}\right)^{2}=\frac{1}{11}+\left(\frac{5}{132}\right)^{2}$$

Square \frac{5}{132} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{66}x+\frac{25}{17424}=\frac{1}{11}+\frac{25}{17424}$$

Add \frac{1}{11} to \frac{25}{17424} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{66}x+\frac{25}{17424}=\frac{1609}{17424}$$

Factor x^{2}+\frac{5}{66}x+\frac{25}{17424}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{132}\right)^{2}=\frac{1609}{17424}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{132}\right)^{2}}=\sqrt{\frac{1609}{17424}}$$

Simplify.

$$x+\frac{5}{132}=\frac{\sqrt{1609}}{132} x+\frac{5}{132}=-\frac{\sqrt{1609}}{132}$$

Subtract \frac{5}{132} from both sides of the equation.

$$x=\frac{\sqrt{1609}-5}{132} x=\frac{-\sqrt{1609}-5}{132}$$