Solve 2x-3/5-5x+18/20*x+6/12=2

$$\frac { 2 x - 3 } { 5 } - \frac { 5 x + 18 } { 20 } \cdot \frac { x + 6 } { 12 } = 2$$

Solve for x (complex solution)

$$x=\frac{24+2\sqrt{771}i}{5}\approx 4.8+11.106754702i$$

$$x=\frac{-2\sqrt{771}i+24}{5}\approx 4.8-11.106754702i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 60, the least common multiple of 5,20,12.

$$12\left(2x-3\right)-3\left(5x+18\right)\times \frac{x+6}{12}=120$$

Use the distributive property to multiply 12 by 2x-3.

$$24x-36-3\left(5x+18\right)\times \frac{x+6}{12}=120$$

Cancel out 12, the greatest common factor in 3 and 12.

$$24x-36-\frac{x+6}{4}\left(5x+18\right)=120$$

Use the distributive property to multiply \frac{x+6}{4} by 5x+18.

$$24x-36-\left(5\times \frac{x+6}{4}x+18\times \frac{x+6}{4}\right)=120$$

Express 5\times \frac{x+6}{4} as a single fraction.

$$24x-36-\left(\frac{5\left(x+6\right)}{4}x+18\times \frac{x+6}{4}\right)=120$$

Express \frac{5\left(x+6\right)}{4}x as a single fraction.

$$24x-36-\left(\frac{5\left(x+6\right)x}{4}+18\times \frac{x+6}{4}\right)=120$$

Express 18\times \frac{x+6}{4} as a single fraction.

$$24x-36-\left(\frac{5\left(x+6\right)x}{4}+\frac{18\left(x+6\right)}{4}\right)=120$$

Since \frac{5\left(x+6\right)x}{4} and \frac{18\left(x+6\right)}{4} have the same denominator, add them by adding their numerators.

$$24x-36-\frac{5\left(x+6\right)x+18\left(x+6\right)}{4}=120$$

Do the multiplications in 5\left(x+6\right)x+18\left(x+6\right).

$$24x-36-\frac{5x^{2}+30x+18x+108}{4}=120$$

Combine like terms in 5x^{2}+30x+18x+108.

$$24x-36-\frac{5x^{2}+48x+108}{4}=120$$

Divide each term of 5x^{2}+48x+108 by 4 to get \frac{5}{4}x^{2}+12x+27.

$$24x-36-\left(\frac{5}{4}x^{2}+12x+27\right)=120$$

To find the opposite of \frac{5}{4}x^{2}+12x+27, find the opposite of each term.

$$24x-36-\frac{5}{4}x^{2}-12x-27=120$$

Combine 24x and -12x to get 12x.

$$12x-36-\frac{5}{4}x^{2}-27=120$$

Subtract 27 from -36 to get -63.

$$12x-63-\frac{5}{4}x^{2}=120$$

Subtract 120 from both sides.

$$12x-63-\frac{5}{4}x^{2}-120=0$$

Subtract 120 from -63 to get -183.

$$12x-183-\frac{5}{4}x^{2}=0$$

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.

$$-\frac{5}{4}x^{2}+12x-183=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{5}{4} for a, 12 for b, and -183 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-\frac{5}{4}\right)\left(-183\right)}}{2\left(-\frac{5}{4}\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-\frac{5}{4}\right)\left(-183\right)}}{2\left(-\frac{5}{4}\right)}$$

Multiply -4 times -\frac{5}{4}.

$$x=\frac{-12±\sqrt{144+5\left(-183\right)}}{2\left(-\frac{5}{4}\right)}$$

Multiply 5 times -183.

$$x=\frac{-12±\sqrt{144-915}}{2\left(-\frac{5}{4}\right)}$$

Add 144 to -915.

$$x=\frac{-12±\sqrt{-771}}{2\left(-\frac{5}{4}\right)}$$

Take the square root of -771.

$$x=\frac{-12±\sqrt{771}i}{2\left(-\frac{5}{4}\right)}$$

Multiply 2 times -\frac{5}{4}.

$$x=\frac{-12±\sqrt{771}i}{-\frac{5}{2}}$$

Now solve the equation x=\frac{-12±\sqrt{771}i}{-\frac{5}{2}} when ± is plus. Add -12 to i\sqrt{771}.

$$x=\frac{-12+\sqrt{771}i}{-\frac{5}{2}}$$

Divide -12+i\sqrt{771} by -\frac{5}{2} by multiplying -12+i\sqrt{771} by the reciprocal of -\frac{5}{2}.

$$x=\frac{-2\sqrt{771}i+24}{5}$$

Now solve the equation x=\frac{-12±\sqrt{771}i}{-\frac{5}{2}} when ± is minus. Subtract i\sqrt{771} from -12.

$$x=\frac{-\sqrt{771}i-12}{-\frac{5}{2}}$$

Divide -12-i\sqrt{771} by -\frac{5}{2} by multiplying -12-i\sqrt{771} by the reciprocal of -\frac{5}{2}.

$$x=\frac{24+2\sqrt{771}i}{5}$$

The equation is now solved.

$$x=\frac{-2\sqrt{771}i+24}{5} x=\frac{24+2\sqrt{771}i}{5}$$

Steps for Completing the Square

Multiply both sides of the equation by 60, the least common multiple of 5,20,12.

$$12\left(2x-3\right)-3\left(5x+18\right)\times \frac{x+6}{12}=120$$

Use the distributive property to multiply 12 by 2x-3.

$$24x-36-3\left(5x+18\right)\times \frac{x+6}{12}=120$$

Cancel out 12, the greatest common factor in 3 and 12.

$$24x-36-\frac{x+6}{4}\left(5x+18\right)=120$$

Use the distributive property to multiply \frac{x+6}{4} by 5x+18.

$$24x-36-\left(5\times \frac{x+6}{4}x+18\times \frac{x+6}{4}\right)=120$$

Express 5\times \frac{x+6}{4} as a single fraction.

$$24x-36-\left(\frac{5\left(x+6\right)}{4}x+18\times \frac{x+6}{4}\right)=120$$

Express \frac{5\left(x+6\right)}{4}x as a single fraction.

$$24x-36-\left(\frac{5\left(x+6\right)x}{4}+18\times \frac{x+6}{4}\right)=120$$

Express 18\times \frac{x+6}{4} as a single fraction.

$$24x-36-\left(\frac{5\left(x+6\right)x}{4}+\frac{18\left(x+6\right)}{4}\right)=120$$

Since \frac{5\left(x+6\right)x}{4} and \frac{18\left(x+6\right)}{4} have the same denominator, add them by adding their numerators.

$$24x-36-\frac{5\left(x+6\right)x+18\left(x+6\right)}{4}=120$$

Do the multiplications in 5\left(x+6\right)x+18\left(x+6\right).

$$24x-36-\frac{5x^{2}+30x+18x+108}{4}=120$$

Combine like terms in 5x^{2}+30x+18x+108.

$$24x-36-\frac{5x^{2}+48x+108}{4}=120$$

Divide each term of 5x^{2}+48x+108 by 4 to get \frac{5}{4}x^{2}+12x+27.

$$24x-36-\left(\frac{5}{4}x^{2}+12x+27\right)=120$$

To find the opposite of \frac{5}{4}x^{2}+12x+27, find the opposite of each term.

$$24x-36-\frac{5}{4}x^{2}-12x-27=120$$

Combine 24x and -12x to get 12x.

$$12x-36-\frac{5}{4}x^{2}-27=120$$

Subtract 27 from -36 to get -63.

$$12x-63-\frac{5}{4}x^{2}=120$$

Add 63 to both sides.

$$12x-\frac{5}{4}x^{2}=120+63$$

Add 120 and 63 to get 183.

$$12x-\frac{5}{4}x^{2}=183$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-\frac{5}{4}x^{2}+12x=183$$

Divide both sides of the equation by -\frac{5}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{5}{4}x^{2}+12x}{-\frac{5}{4}}=\frac{183}{-\frac{5}{4}}$$

Dividing by -\frac{5}{4} undoes the multiplication by -\frac{5}{4}.

$$x^{2}+\frac{12}{-\frac{5}{4}}x=\frac{183}{-\frac{5}{4}}$$

Divide 12 by -\frac{5}{4} by multiplying 12 by the reciprocal of -\frac{5}{4}.

$$x^{2}-\frac{48}{5}x=\frac{183}{-\frac{5}{4}}$$

Divide 183 by -\frac{5}{4} by multiplying 183 by the reciprocal of -\frac{5}{4}.

$$x^{2}-\frac{48}{5}x=-\frac{732}{5}$$

Divide -\frac{48}{5}, the coefficient of the x term, by 2 to get -\frac{24}{5}. Then add the square of -\frac{24}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{48}{5}x+\left(-\frac{24}{5}\right)^{2}=-\frac{732}{5}+\left(-\frac{24}{5}\right)^{2}$$

Square -\frac{24}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{48}{5}x+\frac{576}{25}=-\frac{732}{5}+\frac{576}{25}$$

Add -\frac{732}{5} to \frac{576}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{48}{5}x+\frac{576}{25}=-\frac{3084}{25}$$

Factor x^{2}-\frac{48}{5}x+\frac{576}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{24}{5}\right)^{2}=-\frac{3084}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{24}{5}\right)^{2}}=\sqrt{-\frac{3084}{25}}$$

Simplify.

$$x-\frac{24}{5}=\frac{2\sqrt{771}i}{5} x-\frac{24}{5}=-\frac{2\sqrt{771}i}{5}$$

Add \frac{24}{5} to both sides of the equation.

$$x=\frac{24+2\sqrt{771}i}{5} x=\frac{-2\sqrt{771}i+24}{5}$$