Solve 6x/x+1=6-x/6

$$\frac { 6 x } { x + 1 } = \frac { 6 - x } { 6 }$$

Solve for x

$$x=\frac{\sqrt{985}-31}{2}\approx 0.192354826$$

$$x=\frac{-\sqrt{985}-31}{2}\approx -31.192354826$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by 6\left(x+1\right), the least common multiple of x+1,6.

$$6\times 6x=\left(x+1\right)\left(6-x\right)$$

Multiply 6 and 6 to get 36.

$$36x=\left(x+1\right)\left(6-x\right)$$

Use the distributive property to multiply x+1 by 6-x and combine like terms.

$$36x=5x-x^{2}+6$$

Subtract 5x from both sides.

$$36x-5x=-x^{2}+6$$

Combine 36x and -5x to get 31x.

$$31x=-x^{2}+6$$

Add x^{2} to both sides.

$$31x+x^{2}=6$$

Subtract 6 from both sides.

$$31x+x^{2}-6=0$$

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.

$$x^{2}+31x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 31 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-31±\sqrt{31^{2}-4\left(-6\right)}}{2}$$

Square 31.

$$x=\frac{-31±\sqrt{961-4\left(-6\right)}}{2}$$

Multiply -4 times -6.

$$x=\frac{-31±\sqrt{961+24}}{2}$$

Add 961 to 24.

$$x=\frac{-31±\sqrt{985}}{2}$$

Now solve the equation x=\frac{-31±\sqrt{985}}{2} when ± is plus. Add -31 to \sqrt{985}.

$$x=\frac{\sqrt{985}-31}{2}$$

Now solve the equation x=\frac{-31±\sqrt{985}}{2} when ± is minus. Subtract \sqrt{985} from -31.

$$x=\frac{-\sqrt{985}-31}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{985}-31}{2} x=\frac{-\sqrt{985}-31}{2}$$

Steps for Completing the Square

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by 6\left(x+1\right), the least common multiple of x+1,6.

$$6\times 6x=\left(x+1\right)\left(6-x\right)$$

Multiply 6 and 6 to get 36.

$$36x=\left(x+1\right)\left(6-x\right)$$

Use the distributive property to multiply x+1 by 6-x and combine like terms.

$$36x=5x-x^{2}+6$$

Subtract 5x from both sides.

$$36x-5x=-x^{2}+6$$

Combine 36x and -5x to get 31x.

$$31x=-x^{2}+6$$

Add x^{2} to both sides.

$$31x+x^{2}=6$$

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.

$$x^{2}+31x=6$$

Divide 31, the coefficient of the x term, by 2 to get \frac{31}{2}. Then add the square of \frac{31}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+31x+\left(\frac{31}{2}\right)^{2}=6+\left(\frac{31}{2}\right)^{2}$$

Square \frac{31}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+31x+\frac{961}{4}=6+\frac{961}{4}$$

Add 6 to \frac{961}{4}.

$$x^{2}+31x+\frac{961}{4}=\frac{985}{4}$$

Factor x^{2}+31x+\frac{961}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{31}{2}\right)^{2}=\frac{985}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{31}{2}\right)^{2}}=\sqrt{\frac{985}{4}}$$

Simplify.

$$x+\frac{31}{2}=\frac{\sqrt{985}}{2} x+\frac{31}{2}=-\frac{\sqrt{985}}{2}$$

Subtract \frac{31}{2} from both sides of the equation.

$$x=\frac{\sqrt{985}-31}{2} x=\frac{-\sqrt{985}-31}{2}$$