Solve 3x/x+6=x+3/x

$$\frac{ 3x }{ x+6 } = \frac{ x+3 }{ x }$$

Solve for x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$x=6$$

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 x\left(x+6\right), the least common multiple of x+6,x.

$$x\times 3x=\left(x+6\right)\left(x+3\right)$$

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

$$x^{2}\times 3=\left(x+6\right)\left(x+3\right)$$

Use the distributive property to multiply x+6 by x+3 and combine like terms.

$$x^{2}\times 3=x^{2}+9x+18$$

Subtract x^{2} from both sides.

$$x^{2}\times 3-x^{2}=9x+18$$

Combine x^{2}\times 3 and -x^{2} to get 2x^{2}.

$$2x^{2}=9x+18$$

Subtract 9x from both sides.

$$2x^{2}-9x=18$$

Subtract 18 from both sides.

$$2x^{2}-9x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -9 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 2\left(-18\right)}}{2\times 2}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 2\left(-18\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-9\right)±\sqrt{81-8\left(-18\right)}}{2\times 2}$$

Multiply -8 times -18.

$$x=\frac{-\left(-9\right)±\sqrt{81+144}}{2\times 2}$$

Add 81 to 144.

$$x=\frac{-\left(-9\right)±\sqrt{225}}{2\times 2}$$

Take the square root of 225.

$$x=\frac{-\left(-9\right)±15}{2\times 2}$$

The opposite of -9 is 9.

$$x=\frac{9±15}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{9±15}{4}$$

Now solve the equation x=\frac{9±15}{4} when ± is plus. Add 9 to 15.

$$x=\frac{24}{4}$$

Divide 24 by 4.

$$x=6$$

Now solve the equation x=\frac{9±15}{4} when ± is minus. Subtract 15 from 9.

$$x=-\frac{6}{4}$$

Reduce the fraction \frac{-6}{4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{3}{2}$$

The equation is now solved.

$$x=6 x=-\frac{3}{2}$$

Steps for Completing the Square

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 x\left(x+6\right), the least common multiple of x+6,x.

$$x\times 3x=\left(x+6\right)\left(x+3\right)$$

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

$$x^{2}\times 3=\left(x+6\right)\left(x+3\right)$$

Use the distributive property to multiply x+6 by x+3 and combine like terms.

$$x^{2}\times 3=x^{2}+9x+18$$

Subtract x^{2} from both sides.

$$x^{2}\times 3-x^{2}=9x+18$$

Combine x^{2}\times 3 and -x^{2} to get 2x^{2}.

$$2x^{2}=9x+18$$

Subtract 9x from both sides.

$$2x^{2}-9x=18$$

Divide both sides by 2.

$$\frac{2x^{2}-9x}{2}=\frac{18}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{9}{2}x=\frac{18}{2}$$

Divide 18 by 2.

$$x^{2}-\frac{9}{2}x=9$$

Divide -\frac{9}{2}, the coefficient of the x term, by 2 to get -\frac{9}{4}. Then add the square of -\frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=9+\left(-\frac{9}{4}\right)^{2}$$

Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=9+\frac{81}{16}$$

Add 9 to \frac{81}{16}.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{225}{16}$$

Factor x^{2}-\frac{9}{2}x+\frac{81}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{225}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{225}{16}}$$

Simplify.

$$x-\frac{9}{4}=\frac{15}{4} x-\frac{9}{4}=-\frac{15}{4}$$

Add \frac{9}{4} to both sides of the equation.

$$x=6 x=-\frac{3}{2}$$