Solve 1+7/3-x=3x-18/x^2-3x

$$1 + \frac { 7 } { 3 - x } = \frac { 3 x - 18 } { x ^ { 2 } - 3 x }$$

Solve for x

$$x = \frac{\sqrt{97} + 13}{2} \approx 11.424428901$$

$$x = \frac{13 - \sqrt{97}}{2} \approx 1.575571099$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of 3-x,x^{2}-3x.

$$x\left(x-3\right)-x\times 7=3x-18$$

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

$$x^{2}-3x-x\times 7=3x-18$$

Multiply -1 and 7 to get -7.

$$x^{2}-3x-7x=3x-18$$

Combine -3x and -7x to get -10x.

$$x^{2}-10x=3x-18$$

Subtract 3x from both sides.

$$x^{2}-10x-3x=-18$$

Combine -10x and -3x to get -13x.

$$x^{2}-13x=-18$$

Add 18 to both sides.

$$x^{2}-13x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -13 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 18}}{2}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 18}}{2}$$

Multiply -4 times 18.

$$x=\frac{-\left(-13\right)±\sqrt{169-72}}{2}$$

Add 169 to -72.

$$x=\frac{-\left(-13\right)±\sqrt{97}}{2}$$

The opposite of -13 is 13.

$$x=\frac{13±\sqrt{97}}{2}$$

Now solve the equation x=\frac{13±\sqrt{97}}{2} when ± is plus. Add 13 to \sqrt{97}.

$$x=\frac{\sqrt{97}+13}{2}$$

Now solve the equation x=\frac{13±\sqrt{97}}{2} when ± is minus. Subtract \sqrt{97} from 13.

$$x=\frac{13-\sqrt{97}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{97}+13}{2} x=\frac{13-\sqrt{97}}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of 3-x,x^{2}-3x.

$$x\left(x-3\right)-x\times 7=3x-18$$

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

$$x^{2}-3x-x\times 7=3x-18$$

Multiply -1 and 7 to get -7.

$$x^{2}-3x-7x=3x-18$$

Combine -3x and -7x to get -10x.

$$x^{2}-10x=3x-18$$

Subtract 3x from both sides.

$$x^{2}-10x-3x=-18$$

Combine -10x and -3x to get -13x.

$$x^{2}-13x=-18$$

Divide -13, the coefficient of the x term, by 2 to get -\frac{13}{2}. Then add the square of -\frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=-18+\left(-\frac{13}{2}\right)^{2}$$

Square -\frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-13x+\frac{169}{4}=-18+\frac{169}{4}$$

Add -18 to \frac{169}{4}.

$$x^{2}-13x+\frac{169}{4}=\frac{97}{4}$$

Factor x^{2}-13x+\frac{169}{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}.

$$\left(x-\frac{13}{2}\right)^{2}=\frac{97}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{2}\right)^{2}}=\sqrt{\frac{97}{4}}$$

Simplify.

$$x-\frac{13}{2}=\frac{\sqrt{97}}{2} x-\frac{13}{2}=-\frac{\sqrt{97}}{2}$$

Add \frac{13}{2} to both sides of the equation.

$$x=\frac{\sqrt{97}+13}{2} x=\frac{13-\sqrt{97}}{2}$$