Solve x/6-x=9-3x

$$\frac { x } { 6 - x } = 9 - 3 x$$

Solve for x

$$x = \frac{\sqrt{34} + 14}{3} \approx 6.610317298$$

$$x = \frac{14 - \sqrt{34}}{3} \approx 2.723016035$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by -x+6.

$$x=\left(-x+6\right)\times 9-3x\left(-x+6\right)$$

Use the distributive property to multiply -x+6 by 9.

$$x=-9x+54-3x\left(-x+6\right)$$

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

$$x=-9x+54+3x^{2}-18x$$

Combine -9x and -18x to get -27x.

$$x=-27x+54+3x^{2}$$

Add 27x to both sides.

$$x+27x=54+3x^{2}$$

Combine x and 27x to get 28x.

$$28x=54+3x^{2}$$

Subtract 54 from both sides.

$$28x-54=3x^{2}$$

Subtract 3x^{2} from both sides.

$$28x-54-3x^{2}=0$$

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.

$$-3x^{2}+28x-54=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 28 for b, and -54 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-3\right)\left(-54\right)}}{2\left(-3\right)}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-3\right)\left(-54\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-28±\sqrt{784+12\left(-54\right)}}{2\left(-3\right)}$$

Multiply 12 times -54.

$$x=\frac{-28±\sqrt{784-648}}{2\left(-3\right)}$$

Add 784 to -648.

$$x=\frac{-28±\sqrt{136}}{2\left(-3\right)}$$

Take the square root of 136.

$$x=\frac{-28±2\sqrt{34}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-28±2\sqrt{34}}{-6}$$

Now solve the equation x=\frac{-28±2\sqrt{34}}{-6} when ± is plus. Add -28 to 2\sqrt{34}.

$$x=\frac{2\sqrt{34}-28}{-6}$$

Divide -28+2\sqrt{34} by -6.

$$x=\frac{14-\sqrt{34}}{3}$$

Now solve the equation x=\frac{-28±2\sqrt{34}}{-6} when ± is minus. Subtract 2\sqrt{34} from -28.

$$x=\frac{-2\sqrt{34}-28}{-6}$$

Divide -28-2\sqrt{34} by -6.

$$x=\frac{\sqrt{34}+14}{3}$$

The equation is now solved.

$$x=\frac{14-\sqrt{34}}{3} x=\frac{\sqrt{34}+14}{3}$$

Steps for Completing the Square

Variable x cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by -x+6.

$$x=\left(-x+6\right)\times 9-3x\left(-x+6\right)$$

Use the distributive property to multiply -x+6 by 9.

$$x=-9x+54-3x\left(-x+6\right)$$

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

$$x=-9x+54+3x^{2}-18x$$

Combine -9x and -18x to get -27x.

$$x=-27x+54+3x^{2}$$

Add 27x to both sides.

$$x+27x=54+3x^{2}$$

Combine x and 27x to get 28x.

$$28x=54+3x^{2}$$

Subtract 3x^{2} from both sides.

$$28x-3x^{2}=54$$

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.

$$-3x^{2}+28x=54$$

Divide both sides by -3.

$$\frac{-3x^{2}+28x}{-3}=\frac{54}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{28}{-3}x=\frac{54}{-3}$$

Divide 28 by -3.

$$x^{2}-\frac{28}{3}x=\frac{54}{-3}$$

Divide 54 by -3.

$$x^{2}-\frac{28}{3}x=-18$$

Divide -\frac{28}{3}, the coefficient of the x term, by 2 to get -\frac{14}{3}. Then add the square of -\frac{14}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{28}{3}x+\left(-\frac{14}{3}\right)^{2}=-18+\left(-\frac{14}{3}\right)^{2}$$

Square -\frac{14}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{28}{3}x+\frac{196}{9}=-18+\frac{196}{9}$$

Add -18 to \frac{196}{9}.

$$x^{2}-\frac{28}{3}x+\frac{196}{9}=\frac{34}{9}$$

Factor x^{2}-\frac{28}{3}x+\frac{196}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{14}{3}\right)^{2}=\frac{34}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{14}{3}\right)^{2}}=\sqrt{\frac{34}{9}}$$

Simplify.

$$x-\frac{14}{3}=\frac{\sqrt{34}}{3} x-\frac{14}{3}=-\frac{\sqrt{34}}{3}$$

Add \frac{14}{3} to both sides of the equation.

$$x=\frac{\sqrt{34}+14}{3} x=\frac{14-\sqrt{34}}{3}$$