Solve 12x^2+4x+10=2(3-3x^2)-126=

$$12 x ^ { 2 } + 4 x + 10 = 2 ( 3 - 3 x ^ { 2 } ) - 126 =$$

Solve for x (complex solution)

$$x=\frac{-1+2\sqrt{146}i}{9}\approx -0.111111111+2.685121327i$$

$$x=\frac{-2\sqrt{146}i-1}{9}\approx -0.111111111-2.685121327i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2 by 3-3x^{2}.

$$12x^{2}+4x+10=6-6x^{2}-126$$

Subtract 126 from 6 to get -120.

$$12x^{2}+4x+10=-120-6x^{2}$$

Subtract -120 from both sides.

$$12x^{2}+4x+10-\left(-120\right)=-6x^{2}$$

The opposite of -120 is 120.

$$12x^{2}+4x+10+120=-6x^{2}$$

Add 6x^{2} to both sides.

$$12x^{2}+4x+10+120+6x^{2}=0$$

Add 10 and 120 to get 130.

$$12x^{2}+4x+130+6x^{2}=0$$

Combine 12x^{2} and 6x^{2} to get 18x^{2}.

$$18x^{2}+4x+130=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 18 for a, 4 for b, and 130 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 18\times 130}}{2\times 18}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 18\times 130}}{2\times 18}$$

Multiply -4 times 18.

$$x=\frac{-4±\sqrt{16-72\times 130}}{2\times 18}$$

Multiply -72 times 130.

$$x=\frac{-4±\sqrt{16-9360}}{2\times 18}$$

Add 16 to -9360.

$$x=\frac{-4±\sqrt{-9344}}{2\times 18}$$

Take the square root of -9344.

$$x=\frac{-4±8\sqrt{146}i}{2\times 18}$$

Multiply 2 times 18.

$$x=\frac{-4±8\sqrt{146}i}{36}$$

Now solve the equation x=\frac{-4±8\sqrt{146}i}{36} when ± is plus. Add -4 to 8i\sqrt{146}.

$$x=\frac{-4+8\sqrt{146}i}{36}$$

Divide -4+8i\sqrt{146} by 36.

$$x=\frac{-1+2\sqrt{146}i}{9}$$

Now solve the equation x=\frac{-4±8\sqrt{146}i}{36} when ± is minus. Subtract 8i\sqrt{146} from -4.

$$x=\frac{-8\sqrt{146}i-4}{36}$$

Divide -4-8i\sqrt{146} by 36.

$$x=\frac{-2\sqrt{146}i-1}{9}$$

The equation is now solved.

$$x=\frac{-1+2\sqrt{146}i}{9} x=\frac{-2\sqrt{146}i-1}{9}$$

Steps for Completing the Square

Use the distributive property to multiply 2 by 3-3x^{2}.

$$12x^{2}+4x+10=6-6x^{2}-126$$

Subtract 126 from 6 to get -120.

$$12x^{2}+4x+10=-120-6x^{2}$$

Add 6x^{2} to both sides.

$$12x^{2}+4x+10+6x^{2}=-120$$

Combine 12x^{2} and 6x^{2} to get 18x^{2}.

$$18x^{2}+4x+10=-120$$

Subtract 10 from both sides.

$$18x^{2}+4x=-120-10$$

Subtract 10 from -120 to get -130.

$$18x^{2}+4x=-130$$

Divide both sides by 18.

$$\frac{18x^{2}+4x}{18}=-\frac{130}{18}$$

Dividing by 18 undoes the multiplication by 18.

$$x^{2}+\frac{4}{18}x=-\frac{130}{18}$$

Reduce the fraction \frac{4}{18} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{2}{9}x=-\frac{130}{18}$$

Reduce the fraction \frac{-130}{18} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{2}{9}x=-\frac{65}{9}$$

Divide \frac{2}{9}, the coefficient of the x term, by 2 to get \frac{1}{9}. Then add the square of \frac{1}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{2}{9}x+\left(\frac{1}{9}\right)^{2}=-\frac{65}{9}+\left(\frac{1}{9}\right)^{2}$$

Square \frac{1}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{2}{9}x+\frac{1}{81}=-\frac{65}{9}+\frac{1}{81}$$

Add -\frac{65}{9} to \frac{1}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{2}{9}x+\frac{1}{81}=-\frac{584}{81}$$

Factor x^{2}+\frac{2}{9}x+\frac{1}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{9}\right)^{2}=-\frac{584}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{9}\right)^{2}}=\sqrt{-\frac{584}{81}}$$

Simplify.

$$x+\frac{1}{9}=\frac{2\sqrt{146}i}{9} x+\frac{1}{9}=-\frac{2\sqrt{146}i}{9}$$

Subtract \frac{1}{9} from both sides of the equation.

$$x=\frac{-1+2\sqrt{146}i}{9} x=\frac{-2\sqrt{146}i-1}{9}$$