Solve 6x^2+120x+2070=0

$$6 { x }^{ 2 } +120x+2070 = 0$$

Solve for x (complex solution)

$$x=-10+7\sqrt{5}i\approx -10+15.652475842i$$

$$x=-7\sqrt{5}i-10\approx -10-15.652475842i$$

Steps Using the Quadratic Formula

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.

$$6x^{2}+120x+2070=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 120 for b, and 2070 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-120±\sqrt{120^{2}-4\times 6\times 2070}}{2\times 6}$$

Square 120.

$$x=\frac{-120±\sqrt{14400-4\times 6\times 2070}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-120±\sqrt{14400-24\times 2070}}{2\times 6}$$

Multiply -24 times 2070.

$$x=\frac{-120±\sqrt{14400-49680}}{2\times 6}$$

Add 14400 to -49680.

$$x=\frac{-120±\sqrt{-35280}}{2\times 6}$$

Take the square root of -35280.

$$x=\frac{-120±84\sqrt{5}i}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-120±84\sqrt{5}i}{12}$$

Now solve the equation x=\frac{-120±84\sqrt{5}i}{12} when ± is plus. Add -120 to 84i\sqrt{5}.

$$x=\frac{-120+84\sqrt{5}i}{12}$$

Divide -120+84i\sqrt{5} by 12.

$$x=-10+7\sqrt{5}i$$

Now solve the equation x=\frac{-120±84\sqrt{5}i}{12} when ± is minus. Subtract 84i\sqrt{5} from -120.

$$x=\frac{-84\sqrt{5}i-120}{12}$$

Divide -120-84i\sqrt{5} by 12.

$$x=-7\sqrt{5}i-10$$

The equation is now solved.

$$x=-10+7\sqrt{5}i x=-7\sqrt{5}i-10$$

Steps for Completing the Square

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.

$$6x^{2}+120x+2070=0$$

Subtract 2070 from both sides of the equation.

$$6x^{2}+120x+2070-2070=-2070$$

Subtracting 2070 from itself leaves 0.

$$6x^{2}+120x=-2070$$

Divide both sides by 6.

$$\frac{6x^{2}+120x}{6}=-\frac{2070}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{120}{6}x=-\frac{2070}{6}$$

Divide 120 by 6.

$$x^{2}+20x=-\frac{2070}{6}$$

Divide -2070 by 6.

$$x^{2}+20x=-345$$

Divide 20, the coefficient of the x term, by 2 to get 10. Then add the square of 10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+20x+10^{2}=-345+10^{2}$$

Square 10.

$$x^{2}+20x+100=-345+100$$

Add -345 to 100.

$$x^{2}+20x+100=-245$$

Factor x^{2}+20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+10\right)^{2}=-245$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{-245}$$

Simplify.

$$x+10=7\sqrt{5}i x+10=-7\sqrt{5}i$$

Subtract 10 from both sides of the equation.

$$x=-10+7\sqrt{5}i x=-7\sqrt{5}i-10$$