Solve 3x^2+10/sqrt{2}=5

$$\frac { 3 x ^ { 2 } + 10 } { \sqrt { 2 } } = 5$$

Solve for x (complex solution)

$$x=-\frac{i\sqrt{30-15\sqrt{2}}}{3}\approx -0-0.988084374i$$

$$x=\frac{i\sqrt{30-15\sqrt{2}}}{3}\approx 0.988084374i$$

Steps by Finding Square Root

Rationalize the denominator of \frac{3x^{2}+10}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(3x^{2}+10\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}}=5$$

The square of \sqrt{2} is 2.

$$\frac{\left(3x^{2}+10\right)\sqrt{2}}{2}=5$$

Use the distributive property to multiply 3x^{2}+10 by \sqrt{2}.

$$\frac{3x^{2}\sqrt{2}+10\sqrt{2}}{2}=5$$

Multiply both sides by 2.

$$3x^{2}\sqrt{2}+10\sqrt{2}=5\times 2$$

Multiply 5 and 2 to get 10.

$$3x^{2}\sqrt{2}+10\sqrt{2}=10$$

Subtract 10\sqrt{2} from both sides.

$$3x^{2}\sqrt{2}=10-10\sqrt{2}$$

Dividing by 3\sqrt{2} undoes the multiplication by 3\sqrt{2}.

$$x^{2}=\frac{10-10\sqrt{2}}{3\sqrt{2}}$$

Divide 10-10\sqrt{2} by 3\sqrt{2}.

$$x^{2}=\frac{5\sqrt{2}-10}{3}$$

Take the square root of both sides of the equation.

$$x=\frac{i\sqrt{30-15\sqrt{2}}}{3} x=-\frac{i\sqrt{30-15\sqrt{2}}}{3}$$

The equation is now solved.

$$x=\frac{\sqrt[4]{2}i\sqrt{15\sqrt{2}-15}}{3} x=-\frac{\sqrt[4]{2}i\sqrt{15\sqrt{2}-15}}{3}$$

Steps Using the Quadratic Formula

Rationalize the denominator of \frac{3x^{2}+10}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(3x^{2}+10\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}}=5$$

The square of \sqrt{2} is 2.

$$\frac{\left(3x^{2}+10\right)\sqrt{2}}{2}=5$$

Use the distributive property to multiply 3x^{2}+10 by \sqrt{2}.

$$\frac{3x^{2}\sqrt{2}+10\sqrt{2}}{2}=5$$

Subtract 5 from both sides.

$$\frac{3x^{2}\sqrt{2}+10\sqrt{2}}{2}-5=0$$

Multiply both sides of the equation by 2.

$$3x^{2}\sqrt{2}+10\sqrt{2}-10=0$$

Reorder the terms.

$$3\sqrt{2}x^{2}+10\sqrt{2}-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3\sqrt{2} for a, 0 for b, and 10\sqrt{2}-10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 3\sqrt{2}\left(10\sqrt{2}-10\right)}}{2\times 3\sqrt{2}}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 3\sqrt{2}\left(10\sqrt{2}-10\right)}}{2\times 3\sqrt{2}}$$

Multiply -4 times 3\sqrt{2}.

$$x=\frac{0±\sqrt{\left(-12\sqrt{2}\right)\left(10\sqrt{2}-10\right)}}{2\times 3\sqrt{2}}$$

Multiply -12\sqrt{2} times 10\sqrt{2}-10.

$$x=\frac{0±\sqrt{120\sqrt{2}-240}}{2\times 3\sqrt{2}}$$

Take the square root of -240+120\sqrt{2}.

$$x=\frac{0±2i\sqrt{60-30\sqrt{2}}}{2\times 3\sqrt{2}}$$

Multiply 2 times 3\sqrt{2}.

$$x=\frac{0±2i\sqrt{60-30\sqrt{2}}}{6\sqrt{2}}$$

Now solve the equation x=\frac{0±2i\sqrt{60-30\sqrt{2}}}{6\sqrt{2}} when ± is plus.

$$x=\frac{\sqrt[4]{2}i\sqrt{15\sqrt{2}-15}}{3}$$

Now solve the equation x=\frac{0±2i\sqrt{60-30\sqrt{2}}}{6\sqrt{2}} when ± is minus.

$$x=-\frac{\sqrt[4]{2}i\sqrt{15\sqrt{2}-15}}{3}$$

The equation is now solved.

$$x=\frac{\sqrt[4]{2}i\sqrt{15\sqrt{2}-15}}{3} x=-\frac{\sqrt[4]{2}i\sqrt{15\sqrt{2}-15}}{3}$$