Solve a^2+2-a/2=sqrt{3}

$$a ^ { 2 } + 2 - \frac { a } { 2 } = \sqrt { 3 }$$

Solve for a

$$a=\frac{1+i\sqrt{31-16\sqrt{3}}}{4}\approx 0.25+0.453265036i$$

$$a=\frac{-i\sqrt{31-16\sqrt{3}}+1}{4}\approx 0.25-0.453265036i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

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

Subtract 2\sqrt{3} from both sides.

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

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.

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

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

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

Multiply -4 times 2.

$$a=\frac{-\left(-1\right)±\sqrt{1-8\left(4-2\sqrt{3}\right)}}{2\times 2}$$

Multiply -8 times 4-2\sqrt{3}.

$$a=\frac{-\left(-1\right)±\sqrt{1+16\sqrt{3}-32}}{2\times 2}$$

Add 1 to -32+16\sqrt{3}.

$$a=\frac{-\left(-1\right)±\sqrt{16\sqrt{3}-31}}{2\times 2}$$

Take the square root of -31+16\sqrt{3}.

$$a=\frac{-\left(-1\right)±i\sqrt{31-16\sqrt{3}}}{2\times 2}$$

The opposite of -1 is 1.

$$a=\frac{1±i\sqrt{31-16\sqrt{3}}}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{1±i\sqrt{31-16\sqrt{3}}}{4}$$

Now solve the equation a=\frac{1±i\sqrt{31-16\sqrt{3}}}{4} when ± is plus. Add 1 to i\sqrt{31-16\sqrt{3}}.

$$a=\frac{1+i\sqrt{31-16\sqrt{3}}}{4}$$

Now solve the equation a=\frac{1±i\sqrt{31-16\sqrt{3}}}{4} when ± is minus. Subtract i\sqrt{31-16\sqrt{3}} from 1.

$$a=\frac{-i\sqrt{31-16\sqrt{3}}+1}{4}$$

The equation is now solved.

$$a=\frac{1+i\sqrt{31-16\sqrt{3}}}{4} a=\frac{-i\sqrt{31-16\sqrt{3}}+1}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

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

Subtract 4 from both sides.

$$2a^{2}-a=2\sqrt{3}-4$$

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide 2\sqrt{3}-4 by 2.

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

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{1}{2}a+\frac{1}{16}=\sqrt{3}-2+\frac{1}{16}$$

Add \sqrt{3}-2 to \frac{1}{16}.

$$a^{2}-\frac{1}{2}a+\frac{1}{16}=\sqrt{3}-\frac{31}{16}$$

Factor a^{2}-\frac{1}{2}a+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{1}{4}\right)^{2}=\sqrt{3}-\frac{31}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{1}{4}\right)^{2}}=\sqrt{\sqrt{3}-\frac{31}{16}}$$

Simplify.

$$a-\frac{1}{4}=\frac{i\sqrt{31-16\sqrt{3}}}{4} a-\frac{1}{4}=-\frac{i\sqrt{31-16\sqrt{3}}}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$a=\frac{1+i\sqrt{31-16\sqrt{3}}}{4} a=\frac{-i\sqrt{31-16\sqrt{3}}+1}{4}$$