Solve 64x^2+24sqrt{5}x+33=0

$$64 x ^ { 2 } + 24 \sqrt { 5 } x + 33 = 0$$

Solve for x (complex solution)

$$x=\frac{-3\sqrt{5}+\sqrt{87}i}{16}\approx -0.419262746+0.582961191i$$

$$x=\frac{-\sqrt{87}i-3\sqrt{5}}{16}\approx -0.419262746-0.582961191i$$

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.

$$64x^{2}+24\sqrt{5}x+33=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 64 for a, 24\sqrt{5} for b, and 33 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24\sqrt{5}±\sqrt{\left(24\sqrt{5}\right)^{2}-4\times 64\times 33}}{2\times 64}$$

Square 24\sqrt{5}.

$$x=\frac{-24\sqrt{5}±\sqrt{2880-4\times 64\times 33}}{2\times 64}$$

Multiply -4 times 64.

$$x=\frac{-24\sqrt{5}±\sqrt{2880-256\times 33}}{2\times 64}$$

Multiply -256 times 33.

$$x=\frac{-24\sqrt{5}±\sqrt{2880-8448}}{2\times 64}$$

Add 2880 to -8448.

$$x=\frac{-24\sqrt{5}±\sqrt{-5568}}{2\times 64}$$

Take the square root of -5568.

$$x=\frac{-24\sqrt{5}±8\sqrt{87}i}{2\times 64}$$

Multiply 2 times 64.

$$x=\frac{-24\sqrt{5}±8\sqrt{87}i}{128}$$

Now solve the equation x=\frac{-24\sqrt{5}±8\sqrt{87}i}{128} when ± is plus. Add -24\sqrt{5} to 8i\sqrt{87}.

$$x=\frac{-24\sqrt{5}+8\sqrt{87}i}{128}$$

Divide -24\sqrt{5}+8i\sqrt{87} by 128.

$$x=\frac{-3\sqrt{5}+\sqrt{87}i}{16}$$

Now solve the equation x=\frac{-24\sqrt{5}±8\sqrt{87}i}{128} when ± is minus. Subtract 8i\sqrt{87} from -24\sqrt{5}.

$$x=\frac{-8\sqrt{87}i-24\sqrt{5}}{128}$$

Divide -24\sqrt{5}-8i\sqrt{87} by 128.

$$x=\frac{-\sqrt{87}i-3\sqrt{5}}{16}$$

The equation is now solved.

$$x=\frac{-3\sqrt{5}+\sqrt{87}i}{16} x=\frac{-\sqrt{87}i-3\sqrt{5}}{16}$$

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.

$$64x^{2}+24\sqrt{5}x+33=0$$

Subtract 33 from both sides of the equation.

$$64x^{2}+24\sqrt{5}x+33-33=-33$$

Subtracting 33 from itself leaves 0.

$$64x^{2}+24\sqrt{5}x=-33$$

Divide both sides by 64.

$$\frac{64x^{2}+24\sqrt{5}x}{64}=-\frac{33}{64}$$

Dividing by 64 undoes the multiplication by 64.

$$x^{2}+\frac{24\sqrt{5}}{64}x=-\frac{33}{64}$$

Divide 24\sqrt{5} by 64.

$$x^{2}+\frac{3\sqrt{5}}{8}x=-\frac{33}{64}$$

Divide \frac{3\sqrt{5}}{8}, the coefficient of the x term, by 2 to get \frac{3\sqrt{5}}{16}. Then add the square of \frac{3\sqrt{5}}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3\sqrt{5}}{8}x+\left(\frac{3\sqrt{5}}{16}\right)^{2}=-\frac{33}{64}+\left(\frac{3\sqrt{5}}{16}\right)^{2}$$

Square \frac{3\sqrt{5}}{16}.

$$x^{2}+\frac{3\sqrt{5}}{8}x+\frac{45}{256}=-\frac{33}{64}+\frac{45}{256}$$

Add -\frac{33}{64} to \frac{45}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3\sqrt{5}}{8}x+\frac{45}{256}=-\frac{87}{256}$$

Factor x^{2}+\frac{3\sqrt{5}}{8}x+\frac{45}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3\sqrt{5}}{16}\right)^{2}=-\frac{87}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3\sqrt{5}}{16}\right)^{2}}=\sqrt{-\frac{87}{256}}$$

Simplify.

$$x+\frac{3\sqrt{5}}{16}=\frac{\sqrt{87}i}{16} x+\frac{3\sqrt{5}}{16}=-\frac{\sqrt{87}i}{16}$$

Subtract \frac{3\sqrt{5}}{16} from both sides of the equation.

$$x=\frac{-3\sqrt{5}+\sqrt{87}i}{16} x=\frac{-\sqrt{87}i-3\sqrt{5}}{16}$$