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

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

Solve for x (complex solution)

$$x=-\frac{\sqrt[4]{500}}{5}\approx -0.945741609$$

$$x=\frac{\sqrt[4]{500}}{5}\approx 0.945741609$$

Steps by Finding Square Root

Combine 2x^{2} and 3x^{2} to get 5x^{2}.

$$5x^{2}=2\sqrt{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}=\frac{2\sqrt{5}}{5}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{2}}{\sqrt[4]{5}} x=-\frac{\sqrt{2}}{\sqrt[4]{5}}$$

Steps Using the Quadratic Formula

Combine 2x^{2} and 3x^{2} to get 5x^{2}.

$$5x^{2}=2\sqrt{5}$$

Subtract 2\sqrt{5} from both sides.

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

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

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

Square 0.

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

Multiply -4 times 5.

$$x=\frac{0±\sqrt{-20\left(-2\sqrt{5}\right)}}{2\times 5}$$

Multiply -20 times -2\sqrt{5}.

$$x=\frac{0±\sqrt{40\sqrt{5}}}{2\times 5}$$

Take the square root of 40\sqrt{5}.

$$x=\frac{0±2\sqrt{2}\times 5^{\frac{3}{4}}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{0±2\sqrt{2}\times 5^{\frac{3}{4}}}{10}$$

Now solve the equation x=\frac{0±2\sqrt{2}\times 5^{\frac{3}{4}}}{10} when ± is plus.

$$x=\frac{\sqrt{2}}{\sqrt[4]{5}}$$

Now solve the equation x=\frac{0±2\sqrt{2}\times 5^{\frac{3}{4}}}{10} when ± is minus.

$$x=-\frac{\sqrt{2}\times 5^{\frac{3}{4}}}{5}$$

The equation is now solved.

$$x=\frac{\sqrt{2}}{\sqrt[4]{5}} x=-\frac{\sqrt{2}\times 5^{\frac{3}{4}}}{5}$$

Solve for x

$$x=\frac{\sqrt{2}\times 5^{\frac{3}{4}}}{5}\approx 0.945741609$$

$$x=-\frac{\sqrt{2}\times 5^{\frac{3}{4}}}{5}\approx -0.945741609$$