Solve {left(x+2/-sqrt{2}right)}^2=2x+3

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

Solve for x

$$x=\sqrt{2}\approx 1.414213562$$

$$x=-\sqrt{2}\approx -1.414213562$$

Steps by Finding Square Root

To raise \frac{x+2}{-\sqrt{2}} to a power, raise both numerator and denominator to the power and then divide.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

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

Calculate -\sqrt{2} to the power of 2 and get \left(\sqrt{2}\right)^{2}.

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

The square of \sqrt{2} is 2.

$$\frac{x^{2}+4x+4}{2}=2x+3$$

Divide each term of x^{2}+4x+4 by 2 to get \frac{1}{2}x^{2}+2x+2.

$$\frac{1}{2}x^{2}+2x+2=2x+3$$

Subtract 2x from both sides.

$$\frac{1}{2}x^{2}+2x+2-2x=3$$

Combine 2x and -2x to get 0.

$$\frac{1}{2}x^{2}+2=3$$

Subtract 2 from both sides.

$$\frac{1}{2}x^{2}=3-2$$

Subtract 2 from 3 to get 1.

$$\frac{1}{2}x^{2}=1$$

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$x^{2}=1\times 2$$

Multiply 1 and 2 to get 2.

$$x^{2}=2$$

Take the square root of both sides of the equation.

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

Steps Using the Quadratic Formula

To raise \frac{x+2}{-\sqrt{2}} to a power, raise both numerator and denominator to the power and then divide.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

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

Calculate -\sqrt{2} to the power of 2 and get \left(\sqrt{2}\right)^{2}.

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

The square of \sqrt{2} is 2.

$$\frac{x^{2}+4x+4}{2}=2x+3$$

Divide each term of x^{2}+4x+4 by 2 to get \frac{1}{2}x^{2}+2x+2.

$$\frac{1}{2}x^{2}+2x+2=2x+3$$

Subtract 2x from both sides.

$$\frac{1}{2}x^{2}+2x+2-2x=3$$

Combine 2x and -2x to get 0.

$$\frac{1}{2}x^{2}+2=3$$

Subtract 3 from both sides.

$$\frac{1}{2}x^{2}+2-3=0$$

Subtract 3 from 2 to get -1.

$$\frac{1}{2}x^{2}-1=0$$

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

$$x=\frac{0±\sqrt{0^{2}-4\times \frac{1}{2}\left(-1\right)}}{2\times \frac{1}{2}}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times \frac{1}{2}\left(-1\right)}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$x=\frac{0±\sqrt{-2\left(-1\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -1.

$$x=\frac{0±\sqrt{2}}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$x=\frac{0±\sqrt{2}}{1}$$

Now solve the equation x=\frac{0±\sqrt{2}}{1} when ± is plus.

$$x=\sqrt{2}$$

Now solve the equation x=\frac{0±\sqrt{2}}{1} when ± is minus.

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

The equation is now solved.

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