Solve 85/x+3-9=-x

$$\frac { 85 } { x + 3 } - 9 = - x$$

Solve for x (complex solution)

$$x=3+7i$$

$$x=3-7i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by x+3.

$$85+\left(x+3\right)\left(-9\right)=-\left(x+3\right)x$$

Use the distributive property to multiply x+3 by -9.

$$85-9x-27=-\left(x+3\right)x$$

Subtract 27 from 85 to get 58.

$$58-9x=-\left(x+3\right)x$$

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

$$58-9x=-\left(x^{2}+3x\right)$$

To find the opposite of x^{2}+3x, find the opposite of each term.

$$58-9x=-x^{2}-3x$$

Add x^{2} to both sides.

$$58-9x+x^{2}=-3x$$

Add 3x to both sides.

$$58-9x+x^{2}+3x=0$$

Combine -9x and 3x to get -6x.

$$58-6x+x^{2}=0$$

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.

$$x^{2}-6x+58=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and 58 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 58}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 58}}{2}$$

Multiply -4 times 58.

$$x=\frac{-\left(-6\right)±\sqrt{36-232}}{2}$$

Add 36 to -232.

$$x=\frac{-\left(-6\right)±\sqrt{-196}}{2}$$

Take the square root of -196.

$$x=\frac{-\left(-6\right)±14i}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±14i}{2}$$

Now solve the equation x=\frac{6±14i}{2} when ± is plus. Add 6 to 14i.

$$x=\frac{6+14i}{2}$$

Divide 6+14i by 2.

$$x=3+7i$$

Now solve the equation x=\frac{6±14i}{2} when ± is minus. Subtract 14i from 6.

$$x=\frac{6-14i}{2}$$

Divide 6-14i by 2.

$$x=3-7i$$

The equation is now solved.

$$x=3+7i x=3-7i$$

Steps for Completing the Square

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by x+3.

$$85+\left(x+3\right)\left(-9\right)=-\left(x+3\right)x$$

Use the distributive property to multiply x+3 by -9.

$$85-9x-27=-\left(x+3\right)x$$

Subtract 27 from 85 to get 58.

$$58-9x=-\left(x+3\right)x$$

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

$$58-9x=-\left(x^{2}+3x\right)$$

To find the opposite of x^{2}+3x, find the opposite of each term.

$$58-9x=-x^{2}-3x$$

Add x^{2} to both sides.

$$58-9x+x^{2}=-3x$$

Add 3x to both sides.

$$58-9x+x^{2}+3x=0$$

Combine -9x and 3x to get -6x.

$$58-6x+x^{2}=0$$

Subtract 58 from both sides. Anything subtracted from zero gives its negation.

$$-6x+x^{2}=-58$$

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.

$$x^{2}-6x=-58$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-58+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-58+9$$

Add -58 to 9.

$$x^{2}-6x+9=-49$$

Factor x^{2}-6x+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}.

$$\left(x-3\right)^{2}=-49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{-49}$$

Simplify.

$$x-3=7i x-3=-7i$$

Add 3 to both sides of the equation.

$$x=3+7i x=3-7i$$