Solve x^2+7x+9-x^2=-3x^2

$${ x }^{ 2 } +7x+9- { x }^{ 2 } =-3 { x }^{ 2 }$$

Solve for x (complex solution)

$$x=\frac{-7+\sqrt{59}i}{6}\approx -1.166666667+1.280190958i$$

$$x=\frac{-\sqrt{59}i-7}{6}\approx -1.166666667-1.280190958i$$

Steps Using the Quadratic Formula

Combine x^{2} and -x^{2} to get 0.

$$7x+9=-3x^{2}$$

Add 3x^{2} to both sides.

$$7x+9+3x^{2}=0$$

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.

$$3x^{2}+7x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 7 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 3\times 9}}{2\times 3}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 3\times 9}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-7±\sqrt{49-12\times 9}}{2\times 3}$$

Multiply -12 times 9.

$$x=\frac{-7±\sqrt{49-108}}{2\times 3}$$

Add 49 to -108.

$$x=\frac{-7±\sqrt{-59}}{2\times 3}$$

Take the square root of -59.

$$x=\frac{-7±\sqrt{59}i}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-7±\sqrt{59}i}{6}$$

Now solve the equation x=\frac{-7±\sqrt{59}i}{6} when ± is plus. Add -7 to i\sqrt{59}.

$$x=\frac{-7+\sqrt{59}i}{6}$$

Now solve the equation x=\frac{-7±\sqrt{59}i}{6} when ± is minus. Subtract i\sqrt{59} from -7.

$$x=\frac{-\sqrt{59}i-7}{6}$$

The equation is now solved.

$$x=\frac{-7+\sqrt{59}i}{6} x=\frac{-\sqrt{59}i-7}{6}$$

Steps for Completing the Square

Combine x^{2} and -x^{2} to get 0.

$$7x+9=-3x^{2}$$

Add 3x^{2} to both sides.

$$7x+9+3x^{2}=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$7x+3x^{2}=-9$$

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.

$$3x^{2}+7x=-9$$

Divide both sides by 3.

$$\frac{3x^{2}+7x}{3}=-\frac{9}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{7}{3}x=-\frac{9}{3}$$

Divide -9 by 3.

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

Divide \frac{7}{3}, the coefficient of the x term, by 2 to get \frac{7}{6}. Then add the square of \frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{3}x+\left(\frac{7}{6}\right)^{2}=-3+\left(\frac{7}{6}\right)^{2}$$

Square \frac{7}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=-3+\frac{49}{36}$$

Add -3 to \frac{49}{36}.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=-\frac{59}{36}$$

Factor x^{2}+\frac{7}{3}x+\frac{49}{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}.

$$\left(x+\frac{7}{6}\right)^{2}=-\frac{59}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{6}\right)^{2}}=\sqrt{-\frac{59}{36}}$$

Simplify.

$$x+\frac{7}{6}=\frac{\sqrt{59}i}{6} x+\frac{7}{6}=-\frac{\sqrt{59}i}{6}$$

Subtract \frac{7}{6} from both sides of the equation.

$$x=\frac{-7+\sqrt{59}i}{6} x=\frac{-\sqrt{59}i-7}{6}$$