Solve 84x^2+4sqrt{3}x+3=0

$$84 x ^ { 2 } + 4 \sqrt { 3 } x + 3 = 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42}\approx -0.041239305+0.184427778i$$

$$x=-\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42}\approx -0.041239305-0.184427778i$$

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.

$$84x^{2}+4\sqrt{3}x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 84 for a, 4\sqrt{3} for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4\sqrt{3}±\sqrt{\left(4\sqrt{3}\right)^{2}-4\times 84\times 3}}{2\times 84}$$

Square 4\sqrt{3}.

$$x=\frac{-4\sqrt{3}±\sqrt{48-4\times 84\times 3}}{2\times 84}$$

Multiply -4 times 84.

$$x=\frac{-4\sqrt{3}±\sqrt{48-336\times 3}}{2\times 84}$$

Multiply -336 times 3.

$$x=\frac{-4\sqrt{3}±\sqrt{48-1008}}{2\times 84}$$

Add 48 to -1008.

$$x=\frac{-4\sqrt{3}±\sqrt{-960}}{2\times 84}$$

Take the square root of -960.

$$x=\frac{-4\sqrt{3}±8\sqrt{15}i}{2\times 84}$$

Multiply 2 times 84.

$$x=\frac{-4\sqrt{3}±8\sqrt{15}i}{168}$$

Now solve the equation x=\frac{-4\sqrt{3}±8\sqrt{15}i}{168} when ± is plus. Add -4\sqrt{3} to 8i\sqrt{15}.

$$x=\frac{-4\sqrt{3}+8\sqrt{15}i}{168}$$

Divide -4\sqrt{3}+8i\sqrt{15} by 168.

$$x=\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42}$$

Now solve the equation x=\frac{-4\sqrt{3}±8\sqrt{15}i}{168} when ± is minus. Subtract 8i\sqrt{15} from -4\sqrt{3}.

$$x=\frac{-8\sqrt{15}i-4\sqrt{3}}{168}$$

Divide -4\sqrt{3}-8i\sqrt{15} by 168.

$$x=-\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42}$$

The equation is now solved.

$$x=\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42} x=-\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42}$$

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.

$$84x^{2}+4\sqrt{3}x+3=0$$

Subtract 3 from both sides of the equation.

$$84x^{2}+4\sqrt{3}x+3-3=-3$$

Subtracting 3 from itself leaves 0.

$$84x^{2}+4\sqrt{3}x=-3$$

Divide both sides by 84.

$$\frac{84x^{2}+4\sqrt{3}x}{84}=-\frac{3}{84}$$

Dividing by 84 undoes the multiplication by 84.

$$x^{2}+\frac{4\sqrt{3}}{84}x=-\frac{3}{84}$$

Divide 4\sqrt{3} by 84.

$$x^{2}+\frac{\sqrt{3}}{21}x=-\frac{3}{84}$$

Reduce the fraction \frac{-3}{84} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{\sqrt{3}}{21}x=-\frac{1}{28}$$

Divide \frac{\sqrt{3}}{21}, the coefficient of the x term, by 2 to get \frac{\sqrt{3}}{42}. Then add the square of \frac{\sqrt{3}}{42} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{\sqrt{3}}{21}x+\left(\frac{\sqrt{3}}{42}\right)^{2}=-\frac{1}{28}+\left(\frac{\sqrt{3}}{42}\right)^{2}$$

Square \frac{\sqrt{3}}{42}.

$$x^{2}+\frac{\sqrt{3}}{21}x+\frac{1}{588}=-\frac{1}{28}+\frac{1}{588}$$

Add -\frac{1}{28} to \frac{1}{588} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{\sqrt{3}}{21}x+\frac{1}{588}=-\frac{5}{147}$$

Factor x^{2}+\frac{\sqrt{3}}{21}x+\frac{1}{588}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{\sqrt{3}}{42}\right)^{2}=-\frac{5}{147}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{\sqrt{3}}{42}\right)^{2}}=\sqrt{-\frac{5}{147}}$$

Simplify.

$$x+\frac{\sqrt{3}}{42}=\frac{\sqrt{15}i}{21} x+\frac{\sqrt{3}}{42}=-\frac{\sqrt{15}i}{21}$$

Subtract \frac{\sqrt{3}}{42} from both sides of the equation.

$$x=\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42} x=-\frac{\sqrt{15}i}{21}-\frac{\sqrt{3}}{42}$$