Solve 0=x^2+12/17x-8/9

$$0 = { x }^{ 2 } + \frac{ 12 }{ 17 } x- \frac{ 8 }{ 9 }$$

Solve for x

$$x=\frac{2\sqrt{659}}{51}-\frac{6}{17}\approx 0.653764522$$

$$x=-\frac{2\sqrt{659}}{51}-\frac{6}{17}\approx -1.359646875$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+\frac{12}{17}x-\frac{8}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, \frac{12}{17} for b, and -\frac{8}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{12}{17}±\sqrt{\left(\frac{12}{17}\right)^{2}-4\left(-\frac{8}{9}\right)}}{2}$$

Square \frac{12}{17} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{12}{17}±\sqrt{\frac{144}{289}-4\left(-\frac{8}{9}\right)}}{2}$$

Multiply -4 times -\frac{8}{9}.

$$x=\frac{-\frac{12}{17}±\sqrt{\frac{144}{289}+\frac{32}{9}}}{2}$$

Add \frac{144}{289} to \frac{32}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{12}{17}±\sqrt{\frac{10544}{2601}}}{2}$$

Take the square root of \frac{10544}{2601}.

$$x=\frac{-\frac{12}{17}±\frac{4\sqrt{659}}{51}}{2}$$

Now solve the equation x=\frac{-\frac{12}{17}±\frac{4\sqrt{659}}{51}}{2} when ± is plus. Add -\frac{12}{17} to \frac{4\sqrt{659}}{51}.

$$x=\frac{\frac{4\sqrt{659}}{51}-\frac{12}{17}}{2}$$

Divide -\frac{12}{17}+\frac{4\sqrt{659}}{51} by 2.

$$x=\frac{2\sqrt{659}}{51}-\frac{6}{17}$$

Now solve the equation x=\frac{-\frac{12}{17}±\frac{4\sqrt{659}}{51}}{2} when ± is minus. Subtract \frac{4\sqrt{659}}{51} from -\frac{12}{17}.

$$x=\frac{-\frac{4\sqrt{659}}{51}-\frac{12}{17}}{2}$$

Divide -\frac{12}{17}-\frac{4\sqrt{659}}{51} by 2.

$$x=-\frac{2\sqrt{659}}{51}-\frac{6}{17}$$

The equation is now solved.

$$x=\frac{2\sqrt{659}}{51}-\frac{6}{17} x=-\frac{2\sqrt{659}}{51}-\frac{6}{17}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+\frac{12}{17}x-\frac{8}{9}=0$$

Add \frac{8}{9} to both sides. Anything plus zero gives itself.

$$x^{2}+\frac{12}{17}x=\frac{8}{9}$$

Divide \frac{12}{17}, the coefficient of the x term, by 2 to get \frac{6}{17}. Then add the square of \frac{6}{17} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{12}{17}x+\left(\frac{6}{17}\right)^{2}=\frac{8}{9}+\left(\frac{6}{17}\right)^{2}$$

Square \frac{6}{17} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{12}{17}x+\frac{36}{289}=\frac{8}{9}+\frac{36}{289}$$

Add \frac{8}{9} to \frac{36}{289} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{12}{17}x+\frac{36}{289}=\frac{2636}{2601}$$

Factor x^{2}+\frac{12}{17}x+\frac{36}{289}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{6}{17}\right)^{2}=\frac{2636}{2601}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{6}{17}\right)^{2}}=\sqrt{\frac{2636}{2601}}$$

Simplify.

$$x+\frac{6}{17}=\frac{2\sqrt{659}}{51} x+\frac{6}{17}=-\frac{2\sqrt{659}}{51}$$

Subtract \frac{6}{17} from both sides of the equation.

$$x=\frac{2\sqrt{659}}{51}-\frac{6}{17} x=-\frac{2\sqrt{659}}{51}-\frac{6}{17}$$