Solve 6x^2+12x-1134=0

$$6 { x }^{ 2 } +12x-1134=0$$

Solve for x (complex solution)

$$x=\sqrt{190}-1\approx 12.784048752$$

$$x=-\left(\sqrt{190}+1\right)\approx -14.784048752$$

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.

$$6x^{2}+12x-1134=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 12 for b, and -1134 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 6\left(-1134\right)}}{2\times 6}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 6\left(-1134\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-12±\sqrt{144-24\left(-1134\right)}}{2\times 6}$$

Multiply -24 times -1134.

$$x=\frac{-12±\sqrt{144+27216}}{2\times 6}$$

Add 144 to 27216.

$$x=\frac{-12±\sqrt{27360}}{2\times 6}$$

Take the square root of 27360.

$$x=\frac{-12±12\sqrt{190}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-12±12\sqrt{190}}{12}$$

Now solve the equation x=\frac{-12±12\sqrt{190}}{12} when ± is plus. Add -12 to 12\sqrt{190}.

$$x=\frac{12\sqrt{190}-12}{12}$$

Divide -12+12\sqrt{190} by 12.

$$x=\sqrt{190}-1$$

Now solve the equation x=\frac{-12±12\sqrt{190}}{12} when ± is minus. Subtract 12\sqrt{190} from -12.

$$x=\frac{-12\sqrt{190}-12}{12}$$

Divide -12-12\sqrt{190} by 12.

$$x=-\sqrt{190}-1$$

The equation is now solved.

$$x=\sqrt{190}-1 x=-\sqrt{190}-1$$

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.

$$6x^{2}+12x-1134=0$$

Add 1134 to both sides of the equation.

$$6x^{2}+12x-1134-\left(-1134\right)=-\left(-1134\right)$$

Subtracting -1134 from itself leaves 0.

$$6x^{2}+12x=-\left(-1134\right)$$

Subtract -1134 from 0.

$$6x^{2}+12x=1134$$

Divide both sides by 6.

$$\frac{6x^{2}+12x}{6}=\frac{1134}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{12}{6}x=\frac{1134}{6}$$

Divide 12 by 6.

$$x^{2}+2x=\frac{1134}{6}$$

Divide 1134 by 6.

$$x^{2}+2x=189$$

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

$$x^{2}+2x+1^{2}=189+1^{2}$$

Square 1.

$$x^{2}+2x+1=189+1$$

Add 189 to 1.

$$x^{2}+2x+1=190$$

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

$$\left(x+1\right)^{2}=190$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{190}$$

Simplify.

$$x+1=\sqrt{190} x+1=-\sqrt{190}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{190}-1 x=-\sqrt{190}-1$$

Solve for x

$$x=\sqrt{190}-1\approx 12.784048752$$

$$x=-\sqrt{190}-1\approx -14.784048752$$

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.

$$6x^{2}+12x-1134=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 12 for b, and -1134 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 6\left(-1134\right)}}{2\times 6}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 6\left(-1134\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-12±\sqrt{144-24\left(-1134\right)}}{2\times 6}$$

Multiply -24 times -1134.

$$x=\frac{-12±\sqrt{144+27216}}{2\times 6}$$

Add 144 to 27216.

$$x=\frac{-12±\sqrt{27360}}{2\times 6}$$

Take the square root of 27360.

$$x=\frac{-12±12\sqrt{190}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-12±12\sqrt{190}}{12}$$

Now solve the equation x=\frac{-12±12\sqrt{190}}{12} when ± is plus. Add -12 to 12\sqrt{190}.

$$x=\frac{12\sqrt{190}-12}{12}$$

Divide -12+12\sqrt{190} by 12.

$$x=\sqrt{190}-1$$

Now solve the equation x=\frac{-12±12\sqrt{190}}{12} when ± is minus. Subtract 12\sqrt{190} from -12.

$$x=\frac{-12\sqrt{190}-12}{12}$$

Divide -12-12\sqrt{190} by 12.

$$x=-\sqrt{190}-1$$

The equation is now solved.

$$x=\sqrt{190}-1 x=-\sqrt{190}-1$$

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.

$$6x^{2}+12x-1134=0$$

Add 1134 to both sides of the equation.

$$6x^{2}+12x-1134-\left(-1134\right)=-\left(-1134\right)$$

Subtracting -1134 from itself leaves 0.

$$6x^{2}+12x=-\left(-1134\right)$$

Subtract -1134 from 0.

$$6x^{2}+12x=1134$$

Divide both sides by 6.

$$\frac{6x^{2}+12x}{6}=\frac{1134}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{12}{6}x=\frac{1134}{6}$$

Divide 12 by 6.

$$x^{2}+2x=\frac{1134}{6}$$

Divide 1134 by 6.

$$x^{2}+2x=189$$

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

$$x^{2}+2x+1^{2}=189+1^{2}$$

Square 1.

$$x^{2}+2x+1=189+1$$

Add 189 to 1.

$$x^{2}+2x+1=190$$

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

$$\left(x+1\right)^{2}=190$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{190}$$

Simplify.

$$x+1=\sqrt{190} x+1=-\sqrt{190}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{190}-1 x=-\sqrt{190}-1$$