Solve a^2+4-1a/3=0

$$a ^ { 2 } + 4 - \frac { 1 a } { 3 } = 0$$

Solve for a

$$a=\frac{1+\sqrt{143}i}{6}\approx 0.166666667+1.993043457i$$

$$a=\frac{-\sqrt{143}i+1}{6}\approx 0.166666667-1.993043457i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 3.

$$3a^{2}+12-a=0$$

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.

$$3a^{2}-a+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -1 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-1\right)±\sqrt{1-4\times 3\times 12}}{2\times 3}$$

Multiply -4 times 3.

$$a=\frac{-\left(-1\right)±\sqrt{1-12\times 12}}{2\times 3}$$

Multiply -12 times 12.

$$a=\frac{-\left(-1\right)±\sqrt{1-144}}{2\times 3}$$

Add 1 to -144.

$$a=\frac{-\left(-1\right)±\sqrt{-143}}{2\times 3}$$

Take the square root of -143.

$$a=\frac{-\left(-1\right)±\sqrt{143}i}{2\times 3}$$

The opposite of -1 is 1.

$$a=\frac{1±\sqrt{143}i}{2\times 3}$$

Multiply 2 times 3.

$$a=\frac{1±\sqrt{143}i}{6}$$

Now solve the equation a=\frac{1±\sqrt{143}i}{6} when ± is plus. Add 1 to i\sqrt{143}.

$$a=\frac{1+\sqrt{143}i}{6}$$

Now solve the equation a=\frac{1±\sqrt{143}i}{6} when ± is minus. Subtract i\sqrt{143} from 1.

$$a=\frac{-\sqrt{143}i+1}{6}$$

The equation is now solved.

$$a=\frac{1+\sqrt{143}i}{6} a=\frac{-\sqrt{143}i+1}{6}$$

Steps for Completing the Square

Multiply both sides of the equation by 3.

$$3a^{2}+12-a=0$$

Subtract 12 from both sides. Anything subtracted from zero gives its negation.

$$3a^{2}-a=-12$$

Divide both sides by 3.

$$\frac{3a^{2}-a}{3}=-\frac{12}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$a^{2}-\frac{1}{3}a=-\frac{12}{3}$$

Divide -12 by 3.

$$a^{2}-\frac{1}{3}a=-4$$

Divide -\frac{1}{3}, the coefficient of the x term, by 2 to get -\frac{1}{6}. Then add the square of -\frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{1}{3}a+\left(-\frac{1}{6}\right)^{2}=-4+\left(-\frac{1}{6}\right)^{2}$$

Square -\frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{1}{3}a+\frac{1}{36}=-4+\frac{1}{36}$$

Add -4 to \frac{1}{36}.

$$a^{2}-\frac{1}{3}a+\frac{1}{36}=-\frac{143}{36}$$

Factor a^{2}-\frac{1}{3}a+\frac{1}{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}.

$$\left(a-\frac{1}{6}\right)^{2}=-\frac{143}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{1}{6}\right)^{2}}=\sqrt{-\frac{143}{36}}$$

Simplify.

$$a-\frac{1}{6}=\frac{\sqrt{143}i}{6} a-\frac{1}{6}=-\frac{\sqrt{143}i}{6}$$

Add \frac{1}{6} to both sides of the equation.

$$a=\frac{1+\sqrt{143}i}{6} a=\frac{-\sqrt{143}i+1}{6}$$