Solve -x-9=58/x+3

$$-x-9= \frac{ 58 }{ x+3 }$$

Solve for x (complex solution)

$$x=-6-7i$$

$$x=-6+7i$$

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.

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

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

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

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

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

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

$$-x^{2}-3x-9x-27=58$$

Combine -3x and -9x to get -12x.

$$-x^{2}-12x-27=58$$

Subtract 58 from both sides.

$$-x^{2}-12x-27-58=0$$

Subtract 58 from -27 to get -85.

$$-x^{2}-12x-85=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -12 for b, and -85 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-1\right)\left(-85\right)}}{2\left(-1\right)}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-1\right)\left(-85\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-12\right)±\sqrt{144+4\left(-85\right)}}{2\left(-1\right)}$$

Multiply 4 times -85.

$$x=\frac{-\left(-12\right)±\sqrt{144-340}}{2\left(-1\right)}$$

Add 144 to -340.

$$x=\frac{-\left(-12\right)±\sqrt{-196}}{2\left(-1\right)}$$

Take the square root of -196.

$$x=\frac{-\left(-12\right)±14i}{2\left(-1\right)}$$

The opposite of -12 is 12.

$$x=\frac{12±14i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{12±14i}{-2}$$

Now solve the equation x=\frac{12±14i}{-2} when ± is plus. Add 12 to 14i.

$$x=\frac{12+14i}{-2}$$

Divide 12+14i by -2.

$$x=-6-7i$$

Now solve the equation x=\frac{12±14i}{-2} when ± is minus. Subtract 14i from 12.

$$x=\frac{12-14i}{-2}$$

Divide 12-14i by -2.

$$x=-6+7i$$

The equation is now solved.

$$x=-6-7i x=-6+7i$$

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.

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

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

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

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

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

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

$$-x^{2}-3x-9x-27=58$$

Combine -3x and -9x to get -12x.

$$-x^{2}-12x-27=58$$

Add 27 to both sides.

$$-x^{2}-12x=58+27$$

Add 58 and 27 to get 85.

$$-x^{2}-12x=85$$

Divide both sides by -1.

$$\frac{-x^{2}-12x}{-1}=\frac{85}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{12}{-1}\right)x=\frac{85}{-1}$$

Divide -12 by -1.

$$x^{2}+12x=\frac{85}{-1}$$

Divide 85 by -1.

$$x^{2}+12x=-85$$

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+12x+6^{2}=-85+6^{2}$$

Square 6.

$$x^{2}+12x+36=-85+36$$

Add -85 to 36.

$$x^{2}+12x+36=-49$$

Factor x^{2}+12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=-49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{-49}$$

Simplify.

$$x+6=7i x+6=-7i$$

Subtract 6 from both sides of the equation.

$$x=-6+7i x=-6-7i$$