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

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

Solve for x (complex solution)

$$x=-\left(\sqrt{2}+1\right)\approx -2.414213562$$

$$x=\sqrt{2}+1\approx 2.414213562$$

Steps by Finding Square Root

The equation is now solved.

$$x=\sqrt{2}+1 x=-\left(\sqrt{2}+1\right)$$

Steps Using the Quadratic Formula

Subtract 3 from both sides.

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

Subtract 2\sqrt{2} from both sides.

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

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.

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

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

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

Square 0.

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

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

$$x=\frac{0±\sqrt{8\sqrt{2}+12}}{2}$$

Take the square root of 12+2^{\frac{7}{2}}.

$$x=\frac{0±\left(2\sqrt{2}+2\right)}{2}$$

Now solve the equation x=\frac{0±\left(2\sqrt{2}+2\right)}{2} when ± is plus.

$$x=\sqrt{2}+1$$

Now solve the equation x=\frac{0±\left(2\sqrt{2}+2\right)}{2} when ± is minus.

$$x=-\sqrt{2}-1$$

The equation is now solved.

$$x=\sqrt{2}+1 x=-\sqrt{2}-1$$

Solve for x

$$x=\sqrt{2}+1\approx 2.414213562$$

$$x=-\sqrt{2}-1\approx -2.414213562$$