Solve 3x+10/x-2[9x-4/3-3(x/2+7x-6/4)]=frac{7x+5}{2}

$$\frac { 3 x + 10 } { x } - 2 [ \frac { 9 x - 4 } { 3 } - 3 ( \frac { x } { 2 } + \frac { 7 x - 6 } { 4 } ) ] = \frac { 7 x + 5 } { 2 }$$

Solve for x (complex solution)

$$x=\frac{35+\sqrt{4535}i}{48}\approx 0.729166667+1.402966846i$$

$$x=\frac{-\sqrt{4535}i+35}{48}\approx 0.729166667-1.402966846i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 12x, the least common multiple of x,3,2,4.

$$12\left(3x+10\right)-2\left(\frac{9x-4}{3}-3\left(\frac{x}{2}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

Use the distributive property to multiply 12 by 3x+10.

$$36x+120-2\left(\frac{9x-4}{3}-3\left(\frac{x}{2}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 4 is 4. Multiply \frac{x}{2} times \frac{2}{2}.

$$36x+120-2\left(\frac{9x-4}{3}-3\left(\frac{2x}{4}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

Since \frac{2x}{4} and \frac{7x-6}{4} have the same denominator, add them by adding their numerators.

$$36x+120-2\left(\frac{9x-4}{3}-3\times \frac{2x+7x-6}{4}\right)\times 12x=6x\left(7x+5\right)$$

Combine like terms in 2x+7x-6.

$$36x+120-2\left(\frac{9x-4}{3}-3\times \frac{9x-6}{4}\right)\times 12x=6x\left(7x+5\right)$$

Express 3\times \frac{9x-6}{4} as a single fraction.

$$36x+120-2\left(\frac{9x-4}{3}-\frac{3\left(9x-6\right)}{4}\right)\times 12x=6x\left(7x+5\right)$$

Use the distributive property to multiply 3 by 9x-6.

$$36x+120-2\left(\frac{9x-4}{3}-\frac{27x-18}{4}\right)\times 12x=6x\left(7x+5\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 4 is 12. Multiply \frac{9x-4}{3} times \frac{4}{4}. Multiply \frac{27x-18}{4} times \frac{3}{3}.

$$36x+120-2\left(\frac{4\left(9x-4\right)}{12}-\frac{3\left(27x-18\right)}{12}\right)\times 12x=6x\left(7x+5\right)$$

Since \frac{4\left(9x-4\right)}{12} and \frac{3\left(27x-18\right)}{12} have the same denominator, subtract them by subtracting their numerators.

$$36x+120-2\times \frac{4\left(9x-4\right)-3\left(27x-18\right)}{12}\times 12x=6x\left(7x+5\right)$$

Do the multiplications in 4\left(9x-4\right)-3\left(27x-18\right).

$$36x+120-2\times \frac{36x-16-81x+54}{12}\times 12x=6x\left(7x+5\right)$$

Combine like terms in 36x-16-81x+54.

$$36x+120-2\times \frac{-45x+38}{12}\times 12x=6x\left(7x+5\right)$$

Multiply 2 and 12 to get 24.

$$36x+120-24\times \frac{-45x+38}{12}x=6x\left(7x+5\right)$$

Cancel out 12, the greatest common factor in 24 and 12.

$$36x+120-2\left(-45x+38\right)x=6x\left(7x+5\right)$$

Use the distributive property to multiply 6x by 7x+5.

$$36x+120-2\left(-45x+38\right)x=42x^{2}+30x$$

Subtract 42x^{2} from both sides.

$$36x+120-2\left(-45x+38\right)x-42x^{2}=30x$$

Subtract 30x from both sides.

$$36x+120-2\left(-45x+38\right)x-42x^{2}-30x=0$$

Use the distributive property to multiply -2 by -45x+38.

$$36x+120+\left(90x-76\right)x-42x^{2}-30x=0$$

Use the distributive property to multiply 90x-76 by x.

$$36x+120+90x^{2}-76x-42x^{2}-30x=0$$

Combine 36x and -76x to get -40x.

$$-40x+120+90x^{2}-42x^{2}-30x=0$$

Combine 90x^{2} and -42x^{2} to get 48x^{2}.

$$-40x+120+48x^{2}-30x=0$$

Combine -40x and -30x to get -70x.

$$-70x+120+48x^{2}=0$$

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.

$$48x^{2}-70x+120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 48 for a, -70 for b, and 120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\times 48\times 120}}{2\times 48}$$

Square -70.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\times 48\times 120}}{2\times 48}$$

Multiply -4 times 48.

$$x=\frac{-\left(-70\right)±\sqrt{4900-192\times 120}}{2\times 48}$$

Multiply -192 times 120.

$$x=\frac{-\left(-70\right)±\sqrt{4900-23040}}{2\times 48}$$

Add 4900 to -23040.

$$x=\frac{-\left(-70\right)±\sqrt{-18140}}{2\times 48}$$

Take the square root of -18140.

$$x=\frac{-\left(-70\right)±2\sqrt{4535}i}{2\times 48}$$

The opposite of -70 is 70.

$$x=\frac{70±2\sqrt{4535}i}{2\times 48}$$

Multiply 2 times 48.

$$x=\frac{70±2\sqrt{4535}i}{96}$$

Now solve the equation x=\frac{70±2\sqrt{4535}i}{96} when ± is plus. Add 70 to 2i\sqrt{4535}.

$$x=\frac{70+2\sqrt{4535}i}{96}$$

Divide 70+2i\sqrt{4535} by 96.

$$x=\frac{35+\sqrt{4535}i}{48}$$

Now solve the equation x=\frac{70±2\sqrt{4535}i}{96} when ± is minus. Subtract 2i\sqrt{4535} from 70.

$$x=\frac{-2\sqrt{4535}i+70}{96}$$

Divide 70-2i\sqrt{4535} by 96.

$$x=\frac{-\sqrt{4535}i+35}{48}$$

The equation is now solved.

$$x=\frac{35+\sqrt{4535}i}{48} x=\frac{-\sqrt{4535}i+35}{48}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 12x, the least common multiple of x,3,2,4.

$$12\left(3x+10\right)-2\left(\frac{9x-4}{3}-3\left(\frac{x}{2}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

Use the distributive property to multiply 12 by 3x+10.

$$36x+120-2\left(\frac{9x-4}{3}-3\left(\frac{x}{2}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 4 is 4. Multiply \frac{x}{2} times \frac{2}{2}.

$$36x+120-2\left(\frac{9x-4}{3}-3\left(\frac{2x}{4}+\frac{7x-6}{4}\right)\right)\times 12x=6x\left(7x+5\right)$$

Since \frac{2x}{4} and \frac{7x-6}{4} have the same denominator, add them by adding their numerators.

$$36x+120-2\left(\frac{9x-4}{3}-3\times \frac{2x+7x-6}{4}\right)\times 12x=6x\left(7x+5\right)$$

Combine like terms in 2x+7x-6.

$$36x+120-2\left(\frac{9x-4}{3}-3\times \frac{9x-6}{4}\right)\times 12x=6x\left(7x+5\right)$$

Express 3\times \frac{9x-6}{4} as a single fraction.

$$36x+120-2\left(\frac{9x-4}{3}-\frac{3\left(9x-6\right)}{4}\right)\times 12x=6x\left(7x+5\right)$$

Use the distributive property to multiply 3 by 9x-6.

$$36x+120-2\left(\frac{9x-4}{3}-\frac{27x-18}{4}\right)\times 12x=6x\left(7x+5\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 4 is 12. Multiply \frac{9x-4}{3} times \frac{4}{4}. Multiply \frac{27x-18}{4} times \frac{3}{3}.

$$36x+120-2\left(\frac{4\left(9x-4\right)}{12}-\frac{3\left(27x-18\right)}{12}\right)\times 12x=6x\left(7x+5\right)$$

Since \frac{4\left(9x-4\right)}{12} and \frac{3\left(27x-18\right)}{12} have the same denominator, subtract them by subtracting their numerators.

$$36x+120-2\times \frac{4\left(9x-4\right)-3\left(27x-18\right)}{12}\times 12x=6x\left(7x+5\right)$$

Do the multiplications in 4\left(9x-4\right)-3\left(27x-18\right).

$$36x+120-2\times \frac{36x-16-81x+54}{12}\times 12x=6x\left(7x+5\right)$$

Combine like terms in 36x-16-81x+54.

$$36x+120-2\times \frac{-45x+38}{12}\times 12x=6x\left(7x+5\right)$$

Multiply 2 and 12 to get 24.

$$36x+120-24\times \frac{-45x+38}{12}x=6x\left(7x+5\right)$$

Cancel out 12, the greatest common factor in 24 and 12.

$$36x+120-2\left(-45x+38\right)x=6x\left(7x+5\right)$$

Use the distributive property to multiply 6x by 7x+5.

$$36x+120-2\left(-45x+38\right)x=42x^{2}+30x$$

Subtract 42x^{2} from both sides.

$$36x+120-2\left(-45x+38\right)x-42x^{2}=30x$$

Subtract 30x from both sides.

$$36x+120-2\left(-45x+38\right)x-42x^{2}-30x=0$$

Use the distributive property to multiply -2 by -45x+38.

$$36x+120+\left(90x-76\right)x-42x^{2}-30x=0$$

Use the distributive property to multiply 90x-76 by x.

$$36x+120+90x^{2}-76x-42x^{2}-30x=0$$

Combine 36x and -76x to get -40x.

$$-40x+120+90x^{2}-42x^{2}-30x=0$$

Combine 90x^{2} and -42x^{2} to get 48x^{2}.

$$-40x+120+48x^{2}-30x=0$$

Combine -40x and -30x to get -70x.

$$-70x+120+48x^{2}=0$$

Subtract 120 from both sides. Anything subtracted from zero gives its negation.

$$-70x+48x^{2}=-120$$

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.

$$48x^{2}-70x=-120$$

Divide both sides by 48.

$$\frac{48x^{2}-70x}{48}=-\frac{120}{48}$$

Dividing by 48 undoes the multiplication by 48.

$$x^{2}+\left(-\frac{70}{48}\right)x=-\frac{120}{48}$$

Reduce the fraction \frac{-70}{48} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{35}{24}x=-\frac{120}{48}$$

Reduce the fraction \frac{-120}{48} to lowest terms by extracting and canceling out 24.

$$x^{2}-\frac{35}{24}x=-\frac{5}{2}$$

Divide -\frac{35}{24}, the coefficient of the x term, by 2 to get -\frac{35}{48}. Then add the square of -\frac{35}{48} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{35}{24}x+\left(-\frac{35}{48}\right)^{2}=-\frac{5}{2}+\left(-\frac{35}{48}\right)^{2}$$

Square -\frac{35}{48} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{35}{24}x+\frac{1225}{2304}=-\frac{5}{2}+\frac{1225}{2304}$$

Add -\frac{5}{2} to \frac{1225}{2304} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{35}{24}x+\frac{1225}{2304}=-\frac{4535}{2304}$$

Factor x^{2}-\frac{35}{24}x+\frac{1225}{2304}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{35}{48}\right)^{2}=-\frac{4535}{2304}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{48}\right)^{2}}=\sqrt{-\frac{4535}{2304}}$$

Simplify.

$$x-\frac{35}{48}=\frac{\sqrt{4535}i}{48} x-\frac{35}{48}=-\frac{\sqrt{4535}i}{48}$$

Add \frac{35}{48} to both sides of the equation.

$$x=\frac{35+\sqrt{4535}i}{48} x=\frac{-\sqrt{4535}i+35}{48}$$