Solve 27/x+5/x+2x/5x=8x/7x+2x

$$\frac { 27 } { x } + \frac { 5 } { x } + \frac { 2 x } { 5 x } = \frac { 8 x } { 7 x } + 2 x$$

Solve for x

$$x = \frac{\sqrt{78569} - 13}{70} \approx 3.818594618$$

$$x=\frac{-\sqrt{78569}-13}{70}\approx -4.190023189$$

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 35x, the least common multiple of x,5x,7x.

$$35\times 27+35\times 5+7\times 2x=5\times 8x+2x\times 35x$$

Do the multiplications.

$$945+175+7\times 2x=5\times 8x+2x\times 35x$$

Add 945 and 175 to get 1120.

$$1120+7\times 2x=5\times 8x+2x\times 35x$$

Multiply 7 and 2 to get 14.

$$1120+14x=5\times 8x+2x\times 35x$$

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

$$1120+14x=5\times 8x+2x^{2}\times 35$$

Multiply 5 and 8 to get 40.

$$1120+14x=40x+2x^{2}\times 35$$

Multiply 2 and 35 to get 70.

$$1120+14x=40x+70x^{2}$$

Subtract 40x from both sides.

$$1120+14x-40x=70x^{2}$$

Combine 14x and -40x to get -26x.

$$1120-26x=70x^{2}$$

Subtract 70x^{2} from both sides.

$$1120-26x-70x^{2}=0$$

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.

$$-70x^{2}-26x+1120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -70 for a, -26 for b, and 1120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\left(-70\right)\times 1120}}{2\left(-70\right)}$$

Square -26.

$$x=\frac{-\left(-26\right)±\sqrt{676-4\left(-70\right)\times 1120}}{2\left(-70\right)}$$

Multiply -4 times -70.

$$x=\frac{-\left(-26\right)±\sqrt{676+280\times 1120}}{2\left(-70\right)}$$

Multiply 280 times 1120.

$$x=\frac{-\left(-26\right)±\sqrt{676+313600}}{2\left(-70\right)}$$

Add 676 to 313600.

$$x=\frac{-\left(-26\right)±\sqrt{314276}}{2\left(-70\right)}$$

Take the square root of 314276.

$$x=\frac{-\left(-26\right)±2\sqrt{78569}}{2\left(-70\right)}$$

The opposite of -26 is 26.

$$x=\frac{26±2\sqrt{78569}}{2\left(-70\right)}$$

Multiply 2 times -70.

$$x=\frac{26±2\sqrt{78569}}{-140}$$

Now solve the equation x=\frac{26±2\sqrt{78569}}{-140} when ± is plus. Add 26 to 2\sqrt{78569}.

$$x=\frac{2\sqrt{78569}+26}{-140}$$

Divide 26+2\sqrt{78569} by -140.

$$x=\frac{-\sqrt{78569}-13}{70}$$

Now solve the equation x=\frac{26±2\sqrt{78569}}{-140} when ± is minus. Subtract 2\sqrt{78569} from 26.

$$x=\frac{26-2\sqrt{78569}}{-140}$$

Divide 26-2\sqrt{78569} by -140.

$$x=\frac{\sqrt{78569}-13}{70}$$

The equation is now solved.

$$x=\frac{-\sqrt{78569}-13}{70} x=\frac{\sqrt{78569}-13}{70}$$

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 35x, the least common multiple of x,5x,7x.

$$35\times 27+35\times 5+7\times 2x=5\times 8x+2x\times 35x$$

Do the multiplications.

$$945+175+7\times 2x=5\times 8x+2x\times 35x$$

Add 945 and 175 to get 1120.

$$1120+7\times 2x=5\times 8x+2x\times 35x$$

Multiply 7 and 2 to get 14.

$$1120+14x=5\times 8x+2x\times 35x$$

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

$$1120+14x=5\times 8x+2x^{2}\times 35$$

Multiply 5 and 8 to get 40.

$$1120+14x=40x+2x^{2}\times 35$$

Multiply 2 and 35 to get 70.

$$1120+14x=40x+70x^{2}$$

Subtract 40x from both sides.

$$1120+14x-40x=70x^{2}$$

Combine 14x and -40x to get -26x.

$$1120-26x=70x^{2}$$

Subtract 70x^{2} from both sides.

$$1120-26x-70x^{2}=0$$

Subtract 1120 from both sides. Anything subtracted from zero gives its negation.

$$-26x-70x^{2}=-1120$$

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.

$$-70x^{2}-26x=-1120$$

Divide both sides by -70.

$$\frac{-70x^{2}-26x}{-70}=-\frac{1120}{-70}$$

Dividing by -70 undoes the multiplication by -70.

$$x^{2}+\left(-\frac{26}{-70}\right)x=-\frac{1120}{-70}$$

Reduce the fraction \frac{-26}{-70} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{13}{35}x=-\frac{1120}{-70}$$

Divide -1120 by -70.

$$x^{2}+\frac{13}{35}x=16$$

Divide \frac{13}{35}, the coefficient of the x term, by 2 to get \frac{13}{70}. Then add the square of \frac{13}{70} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{13}{35}x+\left(\frac{13}{70}\right)^{2}=16+\left(\frac{13}{70}\right)^{2}$$

Square \frac{13}{70} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{13}{35}x+\frac{169}{4900}=16+\frac{169}{4900}$$

Add 16 to \frac{169}{4900}.

$$x^{2}+\frac{13}{35}x+\frac{169}{4900}=\frac{78569}{4900}$$

Factor x^{2}+\frac{13}{35}x+\frac{169}{4900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{13}{70}\right)^{2}=\frac{78569}{4900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{70}\right)^{2}}=\sqrt{\frac{78569}{4900}}$$

Simplify.

$$x+\frac{13}{70}=\frac{\sqrt{78569}}{70} x+\frac{13}{70}=-\frac{\sqrt{78569}}{70}$$

Subtract \frac{13}{70} from both sides of the equation.

$$x=\frac{\sqrt{78569}-13}{70} x=\frac{-\sqrt{78569}-13}{70}$$