Solve 3x^2+8-5sqrt{2}=0

$$3 x ^ { 2 } + 8 - 5 \sqrt { 2 } = 0$$

Solve for x (complex solution)

$$x=-\frac{i\sqrt{24-15\sqrt{2}}}{3}\approx -0-0.556456703i$$

$$x=\frac{i\sqrt{24-15\sqrt{2}}}{3}\approx 0.556456703i$$

Steps by Finding Square Root

Subtract 8 from both sides. Anything subtracted from zero gives its negation.

$$3x^{2}-5\sqrt{2}=-8$$

Add 5\sqrt{2} to both sides.

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

Dividing by 3 undoes the multiplication by 3.

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

Take the square root of both sides of the equation.

$$x=\frac{i\sqrt{24-15\sqrt{2}}}{3} x=-\frac{i\sqrt{24-15\sqrt{2}}}{3}$$

Steps Using the Quadratic Formula

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$3x^{2}+8-5\sqrt{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 0 for b, and 8-5\sqrt{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 3\left(8-5\sqrt{2}\right)}}{2\times 3}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 3\left(8-5\sqrt{2}\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{0±\sqrt{-12\left(8-5\sqrt{2}\right)}}{2\times 3}$$

Multiply -12 times 8-5\sqrt{2}.

$$x=\frac{0±\sqrt{60\sqrt{2}-96}}{2\times 3}$$

Take the square root of -96+60\sqrt{2}.

$$x=\frac{0±2i\sqrt{24-15\sqrt{2}}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{0±2i\sqrt{24-15\sqrt{2}}}{6}$$

Now solve the equation x=\frac{0±2i\sqrt{24-15\sqrt{2}}}{6} when ± is plus.

$$x=\frac{i\sqrt{24-15\sqrt{2}}}{3}$$

Now solve the equation x=\frac{0±2i\sqrt{24-15\sqrt{2}}}{6} when ± is minus.

$$x=-\frac{i\sqrt{24-15\sqrt{2}}}{3}$$

The equation is now solved.

$$x=\frac{i\sqrt{24-15\sqrt{2}}}{3} x=-\frac{i\sqrt{24-15\sqrt{2}}}{3}$$