Solve 6x/3+x^2=-1/2

$$\frac { 6 x } { 3 + x ^ { 2 } } = - \frac { 1 } { 2 }$$

Solve for x

$$x=\sqrt{33}-6\approx -0.255437353$$

$$x=-\sqrt{33}-6\approx -11.744562647$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2\left(x^{2}+3\right), the least common multiple of 3+x^{2},2.

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

Multiply 2 and 6 to get 12.

$$12x=-\left(x^{2}+3\right)$$

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

$$12x=-x^{2}-3$$

Add x^{2} to both sides.

$$12x+x^{2}=-3$$

Add 3 to both sides.

$$12x+x^{2}+3=0$$

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.

$$x^{2}+12x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 12 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 3}}{2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 3}}{2}$$

Multiply -4 times 3.

$$x=\frac{-12±\sqrt{144-12}}{2}$$

Add 144 to -12.

$$x=\frac{-12±\sqrt{132}}{2}$$

Take the square root of 132.

$$x=\frac{-12±2\sqrt{33}}{2}$$

Now solve the equation x=\frac{-12±2\sqrt{33}}{2} when ± is plus. Add -12 to 2\sqrt{33}.

$$x=\frac{2\sqrt{33}-12}{2}$$

Divide -12+2\sqrt{33} by 2.

$$x=\sqrt{33}-6$$

Now solve the equation x=\frac{-12±2\sqrt{33}}{2} when ± is minus. Subtract 2\sqrt{33} from -12.

$$x=\frac{-2\sqrt{33}-12}{2}$$

Divide -12-2\sqrt{33} by 2.

$$x=-\sqrt{33}-6$$

The equation is now solved.

$$x=\sqrt{33}-6 x=-\sqrt{33}-6$$

Steps for Completing the Square

Multiply both sides of the equation by 2\left(x^{2}+3\right), the least common multiple of 3+x^{2},2.

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

Multiply 2 and 6 to get 12.

$$12x=-\left(x^{2}+3\right)$$

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

$$12x=-x^{2}-3$$

Add x^{2} to both sides.

$$12x+x^{2}=-3$$

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.

$$x^{2}+12x=-3$$

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.

$$x^{2}+12x+6^{2}=-3+6^{2}$$

Square 6.

$$x^{2}+12x+36=-3+36$$

Add -3 to 36.

$$x^{2}+12x+36=33$$

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}.

$$\left(x+6\right)^{2}=33$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{33}$$

Simplify.

$$x+6=\sqrt{33} x+6=-\sqrt{33}$$

Subtract 6 from both sides of the equation.

$$x=\sqrt{33}-6 x=-\sqrt{33}-6$$

Solve for x (complex solution)

$$x=\sqrt{33}-6\approx -0.255437353$$

$$x=-\left(\sqrt{33}+6\right)\approx -11.744562647$$