Solve x^2+7=sqrt{27}

$$x ^ { 2 } + 7 = \sqrt { 27 }$$

Solve for x (complex solution)

$$x=-i\sqrt{7-3\sqrt{3}}\approx -0-1.343073928i$$

$$x=i\sqrt{7-3\sqrt{3}}\approx 1.343073928i$$

Steps by Finding Square Root

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$x^{2}+7=3\sqrt{3}$$

Subtract 7 from both sides.

$$x^{2}=3\sqrt{3}-7$$

The equation is now solved.

$$x=i\sqrt{7-3\sqrt{3}} x=-i\sqrt{7-3\sqrt{3}}$$

Steps Using the Quadratic Formula

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$x^{2}+7=3\sqrt{3}$$

Subtract 3\sqrt{3} from both sides.

$$x^{2}+7-3\sqrt{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and 7-3\sqrt{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(7-3\sqrt{3}\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(7-3\sqrt{3}\right)}}{2}$$

Multiply -4 times 7-3\sqrt{3}.

$$x=\frac{0±\sqrt{12\sqrt{3}-28}}{2}$$

Take the square root of -28+12\sqrt{3}.

$$x=\frac{0±2i\sqrt{7-3\sqrt{3}}}{2}$$

Now solve the equation x=\frac{0±2i\sqrt{7-3\sqrt{3}}}{2} when ± is plus.

$$x=i\sqrt{7-3\sqrt{3}}$$

Now solve the equation x=\frac{0±2i\sqrt{7-3\sqrt{3}}}{2} when ± is minus.

$$x=-i\sqrt{7-3\sqrt{3}}$$

The equation is now solved.

$$x=i\sqrt{7-3\sqrt{3}} x=-i\sqrt{7-3\sqrt{3}}$$