Solve 13x-1/24x+2/4-2x-4/3=x-5/6

$$\frac { 13 x - 1 } { 2 } \frac { 4 x + 2 } { 4 } - \frac { 2 x - 4 } { 3 } = \frac { x - 5 } { 6 }$$

Solve for x (complex solution)

$$x=\frac{-23+17\sqrt{23}i}{156}\approx -0.147435897+0.522622666i$$

$$x=\frac{-17\sqrt{23}i-23}{156}\approx -0.147435897-0.522622666i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 12, the least common multiple of 2,4,3,6.

$$6\left(13x-1\right)\times \frac{4x+2}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Express 6\times \frac{4x+2}{4} as a single fraction.

$$\frac{6\left(4x+2\right)}{4}\left(13x-1\right)-4\left(2x-4\right)=2\left(x-5\right)$$

Use the distributive property to multiply \frac{6\left(4x+2\right)}{4} by 13x-1.

$$13\times \frac{6\left(4x+2\right)}{4}x-\frac{6\left(4x+2\right)}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Use the distributive property to multiply 6 by 4x+2.

$$13\times \frac{24x+12}{4}x-\frac{6\left(4x+2\right)}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Express 13\times \frac{24x+12}{4} as a single fraction.

$$\frac{13\left(24x+12\right)}{4}x-\frac{6\left(4x+2\right)}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Express \frac{13\left(24x+12\right)}{4}x as a single fraction.

$$\frac{13\left(24x+12\right)x}{4}-\frac{6\left(4x+2\right)}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Use the distributive property to multiply 6 by 4x+2.

$$\frac{13\left(24x+12\right)x}{4}-\frac{24x+12}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Since \frac{13\left(24x+12\right)x}{4} and \frac{24x+12}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{13\left(24x+12\right)x-\left(24x+12\right)}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Do the multiplications in 13\left(24x+12\right)x-\left(24x+12\right).

$$\frac{312x^{2}+156x-24x-12}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Combine like terms in 312x^{2}+156x-24x-12.

$$\frac{312x^{2}+132x-12}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Use the distributive property to multiply -4 by 2x-4.

$$\frac{312x^{2}+132x-12}{4}-8x+16=2\left(x-5\right)$$

Use the distributive property to multiply 2 by x-5.

$$\frac{312x^{2}+132x-12}{4}-8x+16=2x-10$$

Divide each term of 312x^{2}+132x-12 by 4 to get 78x^{2}+33x-3.

$$78x^{2}+33x-3-8x+16=2x-10$$

Combine 33x and -8x to get 25x.

$$78x^{2}+25x-3+16=2x-10$$

Add -3 and 16 to get 13.

$$78x^{2}+25x+13=2x-10$$

Subtract 2x from both sides.

$$78x^{2}+25x+13-2x=-10$$

Combine 25x and -2x to get 23x.

$$78x^{2}+23x+13=-10$$

Add 10 to both sides.

$$78x^{2}+23x+13+10=0$$

Add 13 and 10 to get 23.

$$78x^{2}+23x+23=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 78 for a, 23 for b, and 23 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-23±\sqrt{23^{2}-4\times 78\times 23}}{2\times 78}$$

Square 23.

$$x=\frac{-23±\sqrt{529-4\times 78\times 23}}{2\times 78}$$

Multiply -4 times 78.

$$x=\frac{-23±\sqrt{529-312\times 23}}{2\times 78}$$

Multiply -312 times 23.

$$x=\frac{-23±\sqrt{529-7176}}{2\times 78}$$

Add 529 to -7176.

$$x=\frac{-23±\sqrt{-6647}}{2\times 78}$$

Take the square root of -6647.

$$x=\frac{-23±17\sqrt{23}i}{2\times 78}$$

Multiply 2 times 78.

$$x=\frac{-23±17\sqrt{23}i}{156}$$

Now solve the equation x=\frac{-23±17\sqrt{23}i}{156} when ± is plus. Add -23 to 17i\sqrt{23}.

$$x=\frac{-23+17\sqrt{23}i}{156}$$

Now solve the equation x=\frac{-23±17\sqrt{23}i}{156} when ± is minus. Subtract 17i\sqrt{23} from -23.

$$x=\frac{-17\sqrt{23}i-23}{156}$$

The equation is now solved.

$$x=\frac{-23+17\sqrt{23}i}{156} x=\frac{-17\sqrt{23}i-23}{156}$$

Steps for Completing the Square

Multiply both sides of the equation by 12, the least common multiple of 2,4,3,6.

$$6\left(13x-1\right)\times \frac{4x+2}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Express 6\times \frac{4x+2}{4} as a single fraction.

$$\frac{6\left(4x+2\right)}{4}\left(13x-1\right)-4\left(2x-4\right)=2\left(x-5\right)$$

Use the distributive property to multiply \frac{6\left(4x+2\right)}{4} by 13x-1.

$$13\times \frac{6\left(4x+2\right)}{4}x-\frac{6\left(4x+2\right)}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Use the distributive property to multiply 6 by 4x+2.

$$13\times \frac{24x+12}{4}x-\frac{6\left(4x+2\right)}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Express 13\times \frac{24x+12}{4} as a single fraction.

$$\frac{13\left(24x+12\right)}{4}x-\frac{6\left(4x+2\right)}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Express \frac{13\left(24x+12\right)}{4}x as a single fraction.

$$\frac{13\left(24x+12\right)x}{4}-\frac{6\left(4x+2\right)}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Use the distributive property to multiply 6 by 4x+2.

$$\frac{13\left(24x+12\right)x}{4}-\frac{24x+12}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Since \frac{13\left(24x+12\right)x}{4} and \frac{24x+12}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{13\left(24x+12\right)x-\left(24x+12\right)}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Do the multiplications in 13\left(24x+12\right)x-\left(24x+12\right).

$$\frac{312x^{2}+156x-24x-12}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Combine like terms in 312x^{2}+156x-24x-12.

$$\frac{312x^{2}+132x-12}{4}-4\left(2x-4\right)=2\left(x-5\right)$$

Use the distributive property to multiply -4 by 2x-4.

$$\frac{312x^{2}+132x-12}{4}-8x+16=2\left(x-5\right)$$

Use the distributive property to multiply 2 by x-5.

$$\frac{312x^{2}+132x-12}{4}-8x+16=2x-10$$

Divide each term of 312x^{2}+132x-12 by 4 to get 78x^{2}+33x-3.

$$78x^{2}+33x-3-8x+16=2x-10$$

Combine 33x and -8x to get 25x.

$$78x^{2}+25x-3+16=2x-10$$

Add -3 and 16 to get 13.

$$78x^{2}+25x+13=2x-10$$

Subtract 2x from both sides.

$$78x^{2}+25x+13-2x=-10$$

Combine 25x and -2x to get 23x.

$$78x^{2}+23x+13=-10$$

Subtract 13 from both sides.

$$78x^{2}+23x=-10-13$$

Subtract 13 from -10 to get -23.

$$78x^{2}+23x=-23$$

Divide both sides by 78.

$$\frac{78x^{2}+23x}{78}=-\frac{23}{78}$$

Dividing by 78 undoes the multiplication by 78.

$$x^{2}+\frac{23}{78}x=-\frac{23}{78}$$

Divide \frac{23}{78}, the coefficient of the x term, by 2 to get \frac{23}{156}. Then add the square of \frac{23}{156} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{23}{78}x+\left(\frac{23}{156}\right)^{2}=-\frac{23}{78}+\left(\frac{23}{156}\right)^{2}$$

Square \frac{23}{156} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{23}{78}x+\frac{529}{24336}=-\frac{23}{78}+\frac{529}{24336}$$

Add -\frac{23}{78} to \frac{529}{24336} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{23}{78}x+\frac{529}{24336}=-\frac{6647}{24336}$$

Factor x^{2}+\frac{23}{78}x+\frac{529}{24336}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{23}{156}\right)^{2}=-\frac{6647}{24336}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{23}{156}\right)^{2}}=\sqrt{-\frac{6647}{24336}}$$

Simplify.

$$x+\frac{23}{156}=\frac{17\sqrt{23}i}{156} x+\frac{23}{156}=-\frac{17\sqrt{23}i}{156}$$

Subtract \frac{23}{156} from both sides of the equation.

$$x=\frac{-23+17\sqrt{23}i}{156} x=\frac{-17\sqrt{23}i-23}{156}$$