Solve 7x-7-5-7x/2=4x-3/5-7+4x/3x

$$7 x - 7 - \frac { 5 - 7 x } { 2 } = \frac { 4 x - 3 } { 5 } - \frac { 7 + 4 x } { 3 x }$$

Solve for x (complex solution)

$$x=\frac{227+\sqrt{29951}i}{582}\approx 0.390034364+0.29736009i$$

$$x=\frac{-\sqrt{29951}i+227}{582}\approx 0.390034364-0.29736009i$$

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 30x, the least common multiple of 2,5,3x.

$$7x\times 30x+30x\left(-7\right)-15x\left(5-7x\right)=6x\left(4x-3\right)-10\left(7+4x\right)$$

Multiply 7 and 30 to get 210.

$$210xx+30x\left(-7\right)-15x\left(5-7x\right)=6x\left(4x-3\right)-10\left(7+4x\right)$$

Multiply x and x to get x^{2}.

$$210x^{2}+30x\left(-7\right)-15x\left(5-7x\right)=6x\left(4x-3\right)-10\left(7+4x\right)$$

Multiply 30 and -7 to get -210.

$$210x^{2}-210x-15x\left(5-7x\right)=6x\left(4x-3\right)-10\left(7+4x\right)$$

Use the distributive property to multiply 15x by 5-7x.

$$210x^{2}-210x-\left(75x-105x^{2}\right)=6x\left(4x-3\right)-10\left(7+4x\right)$$

To find the opposite of 75x-105x^{2}, find the opposite of each term.

$$210x^{2}-210x-75x+105x^{2}=6x\left(4x-3\right)-10\left(7+4x\right)$$

Combine -210x and -75x to get -285x.

$$210x^{2}-285x+105x^{2}=6x\left(4x-3\right)-10\left(7+4x\right)$$

Combine 210x^{2} and 105x^{2} to get 315x^{2}.

$$315x^{2}-285x=6x\left(4x-3\right)-10\left(7+4x\right)$$

Use the distributive property to multiply 6x by 4x-3.

$$315x^{2}-285x=24x^{2}-18x-10\left(7+4x\right)$$

Use the distributive property to multiply -10 by 7+4x.

$$315x^{2}-285x=24x^{2}-18x-70-40x$$

Combine -18x and -40x to get -58x.

$$315x^{2}-285x=24x^{2}-58x-70$$

Subtract 24x^{2} from both sides.

$$315x^{2}-285x-24x^{2}=-58x-70$$

Combine 315x^{2} and -24x^{2} to get 291x^{2}.

$$291x^{2}-285x=-58x-70$$

Add 58x to both sides.

$$291x^{2}-285x+58x=-70$$

Combine -285x and 58x to get -227x.

$$291x^{2}-227x=-70$$

Add 70 to both sides.

$$291x^{2}-227x+70=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 291 for a, -227 for b, and 70 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-227\right)±\sqrt{\left(-227\right)^{2}-4\times 291\times 70}}{2\times 291}$$

Square -227.

$$x=\frac{-\left(-227\right)±\sqrt{51529-4\times 291\times 70}}{2\times 291}$$

Multiply -4 times 291.

$$x=\frac{-\left(-227\right)±\sqrt{51529-1164\times 70}}{2\times 291}$$

Multiply -1164 times 70.

$$x=\frac{-\left(-227\right)±\sqrt{51529-81480}}{2\times 291}$$

Add 51529 to -81480.

$$x=\frac{-\left(-227\right)±\sqrt{-29951}}{2\times 291}$$

Take the square root of -29951.

$$x=\frac{-\left(-227\right)±\sqrt{29951}i}{2\times 291}$$

The opposite of -227 is 227.

$$x=\frac{227±\sqrt{29951}i}{2\times 291}$$

Multiply 2 times 291.

$$x=\frac{227±\sqrt{29951}i}{582}$$

Now solve the equation x=\frac{227±\sqrt{29951}i}{582} when ± is plus. Add 227 to i\sqrt{29951}.

$$x=\frac{227+\sqrt{29951}i}{582}$$

Now solve the equation x=\frac{227±\sqrt{29951}i}{582} when ± is minus. Subtract i\sqrt{29951} from 227.

$$x=\frac{-\sqrt{29951}i+227}{582}$$

The equation is now solved.

$$x=\frac{227+\sqrt{29951}i}{582} x=\frac{-\sqrt{29951}i+227}{582}$$

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 30x, the least common multiple of 2,5,3x.

$$7x\times 30x+30x\left(-7\right)-15x\left(5-7x\right)=6x\left(4x-3\right)-10\left(7+4x\right)$$

Multiply 7 and 30 to get 210.

$$210xx+30x\left(-7\right)-15x\left(5-7x\right)=6x\left(4x-3\right)-10\left(7+4x\right)$$

Multiply x and x to get x^{2}.

$$210x^{2}+30x\left(-7\right)-15x\left(5-7x\right)=6x\left(4x-3\right)-10\left(7+4x\right)$$

Multiply 30 and -7 to get -210.

$$210x^{2}-210x-15x\left(5-7x\right)=6x\left(4x-3\right)-10\left(7+4x\right)$$

Use the distributive property to multiply 15x by 5-7x.

$$210x^{2}-210x-\left(75x-105x^{2}\right)=6x\left(4x-3\right)-10\left(7+4x\right)$$

To find the opposite of 75x-105x^{2}, find the opposite of each term.

$$210x^{2}-210x-75x+105x^{2}=6x\left(4x-3\right)-10\left(7+4x\right)$$

Combine -210x and -75x to get -285x.

$$210x^{2}-285x+105x^{2}=6x\left(4x-3\right)-10\left(7+4x\right)$$

Combine 210x^{2} and 105x^{2} to get 315x^{2}.

$$315x^{2}-285x=6x\left(4x-3\right)-10\left(7+4x\right)$$

Use the distributive property to multiply 6x by 4x-3.

$$315x^{2}-285x=24x^{2}-18x-10\left(7+4x\right)$$

Use the distributive property to multiply -10 by 7+4x.

$$315x^{2}-285x=24x^{2}-18x-70-40x$$

Combine -18x and -40x to get -58x.

$$315x^{2}-285x=24x^{2}-58x-70$$

Subtract 24x^{2} from both sides.

$$315x^{2}-285x-24x^{2}=-58x-70$$

Combine 315x^{2} and -24x^{2} to get 291x^{2}.

$$291x^{2}-285x=-58x-70$$

Add 58x to both sides.

$$291x^{2}-285x+58x=-70$$

Combine -285x and 58x to get -227x.

$$291x^{2}-227x=-70$$

Divide both sides by 291.

$$\frac{291x^{2}-227x}{291}=-\frac{70}{291}$$

Dividing by 291 undoes the multiplication by 291.

$$x^{2}-\frac{227}{291}x=-\frac{70}{291}$$

Divide -\frac{227}{291}, the coefficient of the x term, by 2 to get -\frac{227}{582}. Then add the square of -\frac{227}{582} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{227}{291}x+\left(-\frac{227}{582}\right)^{2}=-\frac{70}{291}+\left(-\frac{227}{582}\right)^{2}$$

Square -\frac{227}{582} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{227}{291}x+\frac{51529}{338724}=-\frac{70}{291}+\frac{51529}{338724}$$

Add -\frac{70}{291} to \frac{51529}{338724} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{227}{291}x+\frac{51529}{338724}=-\frac{29951}{338724}$$

Factor x^{2}-\frac{227}{291}x+\frac{51529}{338724}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{227}{582}\right)^{2}=-\frac{29951}{338724}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{227}{582}\right)^{2}}=\sqrt{-\frac{29951}{338724}}$$

Simplify.

$$x-\frac{227}{582}=\frac{\sqrt{29951}i}{582} x-\frac{227}{582}=-\frac{\sqrt{29951}i}{582}$$

Add \frac{227}{582} to both sides of the equation.

$$x=\frac{227+\sqrt{29951}i}{582} x=\frac{-\sqrt{29951}i+227}{582}$$