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

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

Solve for x

$$x=\frac{6\sqrt{37}}{37}\approx 0.986393924$$

$$x=-\frac{6\sqrt{37}}{37}\approx -0.986393924$$

Steps by Finding Square Root

Variable x cannot be equal to any of the values -6,0 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x+6\right), the least common multiple of x+6,6x.

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

Multiply 6x and 6x to get \left(6x\right)^{2}.

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

Expand \left(6x\right)^{2}.

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

Calculate 6 to the power of 2 and get 36.

$$36x^{2}=\left(x+6\right)\left(6-x\right)$$

Consider \left(x+6\right)\left(6-x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 6.

$$36x^{2}=36-x^{2}$$

Add x^{2} to both sides.

$$36x^{2}+x^{2}=36$$

Combine 36x^{2} and x^{2} to get 37x^{2}.

$$37x^{2}=36$$

Divide both sides by 37.

$$x^{2}=\frac{36}{37}$$

Take the square root of both sides of the equation.

$$x=\frac{6\sqrt{37}}{37} x=-\frac{6\sqrt{37}}{37}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -6,0 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x+6\right), the least common multiple of x+6,6x.

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

Multiply 6x and 6x to get \left(6x\right)^{2}.

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

Expand \left(6x\right)^{2}.

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

Calculate 6 to the power of 2 and get 36.

$$36x^{2}=\left(x+6\right)\left(6-x\right)$$

Consider \left(x+6\right)\left(6-x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 6.

$$36x^{2}=36-x^{2}$$

Subtract 36 from both sides.

$$36x^{2}-36=-x^{2}$$

Add x^{2} to both sides.

$$36x^{2}-36+x^{2}=0$$

Combine 36x^{2} and x^{2} to get 37x^{2}.

$$37x^{2}-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 37 for a, 0 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 37\left(-36\right)}}{2\times 37}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 37\left(-36\right)}}{2\times 37}$$

Multiply -4 times 37.

$$x=\frac{0±\sqrt{-148\left(-36\right)}}{2\times 37}$$

Multiply -148 times -36.

$$x=\frac{0±\sqrt{5328}}{2\times 37}$$

Take the square root of 5328.

$$x=\frac{0±12\sqrt{37}}{2\times 37}$$

Multiply 2 times 37.

$$x=\frac{0±12\sqrt{37}}{74}$$

Now solve the equation x=\frac{0±12\sqrt{37}}{74} when ± is plus.

$$x=\frac{6\sqrt{37}}{37}$$

Now solve the equation x=\frac{0±12\sqrt{37}}{74} when ± is minus.

$$x=-\frac{6\sqrt{37}}{37}$$

The equation is now solved.

$$x=\frac{6\sqrt{37}}{37} x=-\frac{6\sqrt{37}}{37}$$