Solve 34x^2-24x-1/(x+1)(x-1)=0

$$\frac { 34 x ^ { 2 } - 24 x - 1 } { ( x + 1 ) ( x - 1 ) } = 0$$

Solve for x

$$x=\frac{\sqrt{178}}{34}+\frac{6}{17}\approx 0.745343061$$

$$x=-\frac{\sqrt{178}}{34}+\frac{6}{17}\approx -0.039460708$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+1\right).

$$34x^{2}-24x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 34 for a, -24 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 34\left(-1\right)}}{2\times 34}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 34\left(-1\right)}}{2\times 34}$$

Multiply -4 times 34.

$$x=\frac{-\left(-24\right)±\sqrt{576-136\left(-1\right)}}{2\times 34}$$

Multiply -136 times -1.

$$x=\frac{-\left(-24\right)±\sqrt{576+136}}{2\times 34}$$

Add 576 to 136.

$$x=\frac{-\left(-24\right)±\sqrt{712}}{2\times 34}$$

Take the square root of 712.

$$x=\frac{-\left(-24\right)±2\sqrt{178}}{2\times 34}$$

The opposite of -24 is 24.

$$x=\frac{24±2\sqrt{178}}{2\times 34}$$

Multiply 2 times 34.

$$x=\frac{24±2\sqrt{178}}{68}$$

Now solve the equation x=\frac{24±2\sqrt{178}}{68} when ± is plus. Add 24 to 2\sqrt{178}.

$$x=\frac{2\sqrt{178}+24}{68}$$

Divide 24+2\sqrt{178} by 68.

$$x=\frac{\sqrt{178}}{34}+\frac{6}{17}$$

Now solve the equation x=\frac{24±2\sqrt{178}}{68} when ± is minus. Subtract 2\sqrt{178} from 24.

$$x=\frac{24-2\sqrt{178}}{68}$$

Divide 24-2\sqrt{178} by 68.

$$x=-\frac{\sqrt{178}}{34}+\frac{6}{17}$$

The equation is now solved.

$$x=\frac{\sqrt{178}}{34}+\frac{6}{17} x=-\frac{\sqrt{178}}{34}+\frac{6}{17}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+1\right).

$$34x^{2}-24x-1=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$34x^{2}-24x=1$$

Divide both sides by 34.

$$\frac{34x^{2}-24x}{34}=\frac{1}{34}$$

Dividing by 34 undoes the multiplication by 34.

$$x^{2}+\left(-\frac{24}{34}\right)x=\frac{1}{34}$$

Reduce the fraction \frac{-24}{34} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{12}{17}x=\frac{1}{34}$$

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.

$$x^{2}-\frac{12}{17}x+\left(-\frac{6}{17}\right)^{2}=\frac{1}{34}+\left(-\frac{6}{17}\right)^{2}$$

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

$$x^{2}-\frac{12}{17}x+\frac{36}{289}=\frac{1}{34}+\frac{36}{289}$$

Add \frac{1}{34} to \frac{36}{289} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{12}{17}x+\frac{36}{289}=\frac{89}{578}$$

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}.

$$\left(x-\frac{6}{17}\right)^{2}=\frac{89}{578}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{6}{17}\right)^{2}}=\sqrt{\frac{89}{578}}$$

Simplify.

$$x-\frac{6}{17}=\frac{\sqrt{178}}{34} x-\frac{6}{17}=-\frac{\sqrt{178}}{34}$$

Add \frac{6}{17} to both sides of the equation.

$$x=\frac{\sqrt{178}}{34}+\frac{6}{17} x=-\frac{\sqrt{178}}{34}+\frac{6}{17}$$