Solve 72/7-x=x/3

$$7 \frac { 2 } { 7 - x } = \frac { x } { 3 }$$

Solve for x (complex solution)

$$x=\frac{7+\sqrt{119}i}{2}\approx 3.5+5.454356057i$$

$$x=\frac{-\sqrt{119}i+7}{2}\approx 3.5-5.454356057i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 7 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-7\right), the least common multiple of 7-x,3.

$$7\left(-3\right)\times 2=\left(x-7\right)x$$

Multiply 7 and -3 to get -21.

$$-21\times 2=\left(x-7\right)x$$

Multiply -21 and 2 to get -42.

$$-42=\left(x-7\right)x$$

Use the distributive property to multiply x-7 by x.

$$-42=x^{2}-7x$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-7x=-42$$

Add 42 to both sides.

$$x^{2}-7x+42=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -7 for b, and 42 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 42}}{2}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 42}}{2}$$

Multiply -4 times 42.

$$x=\frac{-\left(-7\right)±\sqrt{49-168}}{2}$$

Add 49 to -168.

$$x=\frac{-\left(-7\right)±\sqrt{-119}}{2}$$

Take the square root of -119.

$$x=\frac{-\left(-7\right)±\sqrt{119}i}{2}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{119}i}{2}$$

Now solve the equation x=\frac{7±\sqrt{119}i}{2} when ± is plus. Add 7 to i\sqrt{119}.

$$x=\frac{7+\sqrt{119}i}{2}$$

Now solve the equation x=\frac{7±\sqrt{119}i}{2} when ± is minus. Subtract i\sqrt{119} from 7.

$$x=\frac{-\sqrt{119}i+7}{2}$$

The equation is now solved.

$$x=\frac{7+\sqrt{119}i}{2} x=\frac{-\sqrt{119}i+7}{2}$$

Steps for Completing the Square

Variable x cannot be equal to 7 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-7\right), the least common multiple of 7-x,3.

$$7\left(-3\right)\times 2=\left(x-7\right)x$$

Multiply 7 and -3 to get -21.

$$-21\times 2=\left(x-7\right)x$$

Multiply -21 and 2 to get -42.

$$-42=\left(x-7\right)x$$

Use the distributive property to multiply x-7 by x.

$$-42=x^{2}-7x$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-7x=-42$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-42+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=-42+\frac{49}{4}$$

Add -42 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=-\frac{119}{4}$$

Factor x^{2}-7x+\frac{49}{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}.

$$\left(x-\frac{7}{2}\right)^{2}=-\frac{119}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{-\frac{119}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{\sqrt{119}i}{2} x-\frac{7}{2}=-\frac{\sqrt{119}i}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=\frac{7+\sqrt{119}i}{2} x=\frac{-\sqrt{119}i+7}{2}$$