Solve 8x-2/2x-1-13x-3/1x-2=3

$$\frac{ 8x-2 }{ 2x-1 } - \frac{ 13x-3 }{ 1x-2 } =3$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{14}i}{12}+\frac{1}{3}\approx 0.333333333-0.311804782i$$

$$x=\frac{\sqrt{14}i}{12}+\frac{1}{3}\approx 0.333333333+0.311804782i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values \frac{1}{2},2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(2x-1\right), the least common multiple of 2x-1,1x-2.

$$\left(x-2\right)\left(8x-2\right)-\left(2x-1\right)\left(13x-3\right)=3\left(x-2\right)\left(2x-1\right)$$

Use the distributive property to multiply x-2 by 8x-2 and combine like terms.

$$8x^{2}-18x+4-\left(2x-1\right)\left(13x-3\right)=3\left(x-2\right)\left(2x-1\right)$$

Use the distributive property to multiply 2x-1 by 13x-3 and combine like terms.

$$8x^{2}-18x+4-\left(26x^{2}-19x+3\right)=3\left(x-2\right)\left(2x-1\right)$$

To find the opposite of 26x^{2}-19x+3, find the opposite of each term.

$$8x^{2}-18x+4-26x^{2}+19x-3=3\left(x-2\right)\left(2x-1\right)$$

Combine 8x^{2} and -26x^{2} to get -18x^{2}.

$$-18x^{2}-18x+4+19x-3=3\left(x-2\right)\left(2x-1\right)$$

Combine -18x and 19x to get x.

$$-18x^{2}+x+4-3=3\left(x-2\right)\left(2x-1\right)$$

Subtract 3 from 4 to get 1.

$$-18x^{2}+x+1=3\left(x-2\right)\left(2x-1\right)$$

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

$$-18x^{2}+x+1=\left(3x-6\right)\left(2x-1\right)$$

Use the distributive property to multiply 3x-6 by 2x-1 and combine like terms.

$$-18x^{2}+x+1=6x^{2}-15x+6$$

Subtract 6x^{2} from both sides.

$$-18x^{2}+x+1-6x^{2}=-15x+6$$

Combine -18x^{2} and -6x^{2} to get -24x^{2}.

$$-24x^{2}+x+1=-15x+6$$

Add 15x to both sides.

$$-24x^{2}+x+1+15x=6$$

Combine x and 15x to get 16x.

$$-24x^{2}+16x+1=6$$

Subtract 6 from both sides.

$$-24x^{2}+16x+1-6=0$$

Subtract 6 from 1 to get -5.

$$-24x^{2}+16x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -24 for a, 16 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-24\right)\left(-5\right)}}{2\left(-24\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-24\right)\left(-5\right)}}{2\left(-24\right)}$$

Multiply -4 times -24.

$$x=\frac{-16±\sqrt{256+96\left(-5\right)}}{2\left(-24\right)}$$

Multiply 96 times -5.

$$x=\frac{-16±\sqrt{256-480}}{2\left(-24\right)}$$

Add 256 to -480.

$$x=\frac{-16±\sqrt{-224}}{2\left(-24\right)}$$

Take the square root of -224.

$$x=\frac{-16±4\sqrt{14}i}{2\left(-24\right)}$$

Multiply 2 times -24.

$$x=\frac{-16±4\sqrt{14}i}{-48}$$

Now solve the equation x=\frac{-16±4\sqrt{14}i}{-48} when ± is plus. Add -16 to 4i\sqrt{14}.

$$x=\frac{-16+4\sqrt{14}i}{-48}$$

Divide -16+4i\sqrt{14} by -48.

$$x=-\frac{\sqrt{14}i}{12}+\frac{1}{3}$$

Now solve the equation x=\frac{-16±4\sqrt{14}i}{-48} when ± is minus. Subtract 4i\sqrt{14} from -16.

$$x=\frac{-4\sqrt{14}i-16}{-48}$$

Divide -16-4i\sqrt{14} by -48.

$$x=\frac{\sqrt{14}i}{12}+\frac{1}{3}$$

The equation is now solved.

$$x=-\frac{\sqrt{14}i}{12}+\frac{1}{3} x=\frac{\sqrt{14}i}{12}+\frac{1}{3}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values \frac{1}{2},2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(2x-1\right), the least common multiple of 2x-1,1x-2.

$$\left(x-2\right)\left(8x-2\right)-\left(2x-1\right)\left(13x-3\right)=3\left(x-2\right)\left(2x-1\right)$$

Use the distributive property to multiply x-2 by 8x-2 and combine like terms.

$$8x^{2}-18x+4-\left(2x-1\right)\left(13x-3\right)=3\left(x-2\right)\left(2x-1\right)$$

Use the distributive property to multiply 2x-1 by 13x-3 and combine like terms.

$$8x^{2}-18x+4-\left(26x^{2}-19x+3\right)=3\left(x-2\right)\left(2x-1\right)$$

To find the opposite of 26x^{2}-19x+3, find the opposite of each term.

$$8x^{2}-18x+4-26x^{2}+19x-3=3\left(x-2\right)\left(2x-1\right)$$

Combine 8x^{2} and -26x^{2} to get -18x^{2}.

$$-18x^{2}-18x+4+19x-3=3\left(x-2\right)\left(2x-1\right)$$

Combine -18x and 19x to get x.

$$-18x^{2}+x+4-3=3\left(x-2\right)\left(2x-1\right)$$

Subtract 3 from 4 to get 1.

$$-18x^{2}+x+1=3\left(x-2\right)\left(2x-1\right)$$

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

$$-18x^{2}+x+1=\left(3x-6\right)\left(2x-1\right)$$

Use the distributive property to multiply 3x-6 by 2x-1 and combine like terms.

$$-18x^{2}+x+1=6x^{2}-15x+6$$

Subtract 6x^{2} from both sides.

$$-18x^{2}+x+1-6x^{2}=-15x+6$$

Combine -18x^{2} and -6x^{2} to get -24x^{2}.

$$-24x^{2}+x+1=-15x+6$$

Add 15x to both sides.

$$-24x^{2}+x+1+15x=6$$

Combine x and 15x to get 16x.

$$-24x^{2}+16x+1=6$$

Subtract 1 from both sides.

$$-24x^{2}+16x=6-1$$

Subtract 1 from 6 to get 5.

$$-24x^{2}+16x=5$$

Divide both sides by -24.

$$\frac{-24x^{2}+16x}{-24}=\frac{5}{-24}$$

Dividing by -24 undoes the multiplication by -24.

$$x^{2}+\frac{16}{-24}x=\frac{5}{-24}$$

Reduce the fraction \frac{16}{-24} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{2}{3}x=\frac{5}{-24}$$

Divide 5 by -24.

$$x^{2}-\frac{2}{3}x=-\frac{5}{24}$$

Divide -\frac{2}{3}, the coefficient of the x term, by 2 to get -\frac{1}{3}. Then add the square of -\frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2}{3}x+\left(-\frac{1}{3}\right)^{2}=-\frac{5}{24}+\left(-\frac{1}{3}\right)^{2}$$

Square -\frac{1}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=-\frac{5}{24}+\frac{1}{9}$$

Add -\frac{5}{24} to \frac{1}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=-\frac{7}{72}$$

Factor x^{2}-\frac{2}{3}x+\frac{1}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{3}\right)^{2}=-\frac{7}{72}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{-\frac{7}{72}}$$

Simplify.

$$x-\frac{1}{3}=\frac{\sqrt{14}i}{12} x-\frac{1}{3}=-\frac{\sqrt{14}i}{12}$$

Add \frac{1}{3} to both sides of the equation.

$$x=\frac{\sqrt{14}i}{12}+\frac{1}{3} x=-\frac{\sqrt{14}i}{12}+\frac{1}{3}$$