Solve 2/7-x=x/3

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

Solve for x

$$x=1$$

$$x=6$$

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.

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

Multiply -3 and 2 to get -6.

$$-6=\left(x-7\right)x$$

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

$$-6=x^{2}-7x$$

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

$$x^{2}-7x=-6$$

Add 6 to both sides.

$$x^{2}-7x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -7 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 6}}{2}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 6}}{2}$$

Multiply -4 times 6.

$$x=\frac{-\left(-7\right)±\sqrt{49-24}}{2}$$

Add 49 to -24.

$$x=\frac{-\left(-7\right)±\sqrt{25}}{2}$$

Take the square root of 25.

$$x=\frac{-\left(-7\right)±5}{2}$$

The opposite of -7 is 7.

$$x=\frac{7±5}{2}$$

Now solve the equation x=\frac{7±5}{2} when ± is plus. Add 7 to 5.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

Now solve the equation x=\frac{7±5}{2} when ± is minus. Subtract 5 from 7.

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

Divide 2 by 2.

$$x=1$$

The equation is now solved.

$$x=6 x=1$$

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.

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

Multiply -3 and 2 to get -6.

$$-6=\left(x-7\right)x$$

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

$$-6=x^{2}-7x$$

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

$$x^{2}-7x=-6$$

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.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-6+\left(-\frac{7}{2}\right)^{2}$$

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

$$x^{2}-7x+\frac{49}{4}=-6+\frac{49}{4}$$

Add -6 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{25}{4}$$

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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{5}{2} x-\frac{7}{2}=-\frac{5}{2}$$

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

$$x=6 x=1$$