Solve 2x^2-6/3x-2=13

$$\frac { 2 x ^ { 2 } - 6 } { 3 x - 2 } = 13$$

Solve for x

$$x = \frac{\sqrt{1361} + 39}{4} \approx 18.972933373$$

$$x=\frac{39-\sqrt{1361}}{4}\approx 0.527066627$$

Steps Using the Quadratic Formula

Variable x cannot be equal to \frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by 3x-2.

$$2x^{2}-6=13\left(3x-2\right)$$

Use the distributive property to multiply 13 by 3x-2.

$$2x^{2}-6=39x-26$$

Subtract 39x from both sides.

$$2x^{2}-6-39x=-26$$

Add 26 to both sides.

$$2x^{2}-6-39x+26=0$$

Add -6 and 26 to get 20.

$$2x^{2}+20-39x=0$$

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.

$$2x^{2}-39x+20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -39 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-39\right)±\sqrt{\left(-39\right)^{2}-4\times 2\times 20}}{2\times 2}$$

Square -39.

$$x=\frac{-\left(-39\right)±\sqrt{1521-4\times 2\times 20}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-39\right)±\sqrt{1521-8\times 20}}{2\times 2}$$

Multiply -8 times 20.

$$x=\frac{-\left(-39\right)±\sqrt{1521-160}}{2\times 2}$$

Add 1521 to -160.

$$x=\frac{-\left(-39\right)±\sqrt{1361}}{2\times 2}$$

The opposite of -39 is 39.

$$x=\frac{39±\sqrt{1361}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{39±\sqrt{1361}}{4}$$

Now solve the equation x=\frac{39±\sqrt{1361}}{4} when ± is plus. Add 39 to \sqrt{1361}.

$$x=\frac{\sqrt{1361}+39}{4}$$

Now solve the equation x=\frac{39±\sqrt{1361}}{4} when ± is minus. Subtract \sqrt{1361} from 39.

$$x=\frac{39-\sqrt{1361}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{1361}+39}{4} x=\frac{39-\sqrt{1361}}{4}$$

Steps for Completing the Square

Variable x cannot be equal to \frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by 3x-2.

$$2x^{2}-6=13\left(3x-2\right)$$

Use the distributive property to multiply 13 by 3x-2.

$$2x^{2}-6=39x-26$$

Subtract 39x from both sides.

$$2x^{2}-6-39x=-26$$

Add 6 to both sides.

$$2x^{2}-39x=-26+6$$

Add -26 and 6 to get -20.

$$2x^{2}-39x=-20$$

Divide both sides by 2.

$$\frac{2x^{2}-39x}{2}=-\frac{20}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{39}{2}x=-\frac{20}{2}$$

Divide -20 by 2.

$$x^{2}-\frac{39}{2}x=-10$$

Divide -\frac{39}{2}, the coefficient of the x term, by 2 to get -\frac{39}{4}. Then add the square of -\frac{39}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{39}{2}x+\left(-\frac{39}{4}\right)^{2}=-10+\left(-\frac{39}{4}\right)^{2}$$

Square -\frac{39}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{39}{2}x+\frac{1521}{16}=-10+\frac{1521}{16}$$

Add -10 to \frac{1521}{16}.

$$x^{2}-\frac{39}{2}x+\frac{1521}{16}=\frac{1361}{16}$$

Factor x^{2}-\frac{39}{2}x+\frac{1521}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{39}{4}\right)^{2}=\frac{1361}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{39}{4}\right)^{2}}=\sqrt{\frac{1361}{16}}$$

Simplify.

$$x-\frac{39}{4}=\frac{\sqrt{1361}}{4} x-\frac{39}{4}=-\frac{\sqrt{1361}}{4}$$

Add \frac{39}{4} to both sides of the equation.

$$x=\frac{\sqrt{1361}+39}{4} x=\frac{39-\sqrt{1361}}{4}$$