Solve x^2=(frac{3sqrt{3}}{2})^2+(11/2)^2

$$x ^ { 2 } = ( \frac { 3 \sqrt { 3 } } { 2 } ) ^ { 2 } + ( \frac { 11 } { 2 } ) ^ { 2 }$$

Solve for x

$$x=\sqrt{37}\approx 6.08276253$$

$$x=-\sqrt{37}\approx -6.08276253$$

Steps by Finding Square Root

To raise \frac{3\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$x^{2}=\frac{\left(3\sqrt{3}\right)^{2}}{2^{2}}+\left(\frac{11}{2}\right)^{2}$$

Calculate \frac{11}{2} to the power of 2 and get \frac{121}{4}.

$$x^{2}=\frac{\left(3\sqrt{3}\right)^{2}}{2^{2}}+\frac{121}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Expand 2^{2}.

$$x^{2}=\frac{\left(3\sqrt{3}\right)^{2}}{4}+\frac{121}{4}$$

Since \frac{\left(3\sqrt{3}\right)^{2}}{4} and \frac{121}{4} have the same denominator, add them by adding their numerators.

$$x^{2}=\frac{\left(3\sqrt{3}\right)^{2}+121}{4}$$

Expand \left(3\sqrt{3}\right)^{2}.

$$x^{2}=\frac{3^{2}\left(\sqrt{3}\right)^{2}+121}{4}$$

Calculate 3 to the power of 2 and get 9.

$$x^{2}=\frac{9\left(\sqrt{3}\right)^{2}+121}{4}$$

The square of \sqrt{3} is 3.

$$x^{2}=\frac{9\times 3+121}{4}$$

Multiply 9 and 3 to get 27.

$$x^{2}=\frac{27+121}{4}$$

Add 27 and 121 to get 148.

$$x^{2}=\frac{148}{4}$$

Divide 148 by 4 to get 37.

$$x^{2}=37$$

Take the square root of both sides of the equation.

$$x=\sqrt{37} x=-\sqrt{37}$$

Steps Using the Quadratic Formula

To raise \frac{3\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$x^{2}=\frac{\left(3\sqrt{3}\right)^{2}}{2^{2}}+\left(\frac{11}{2}\right)^{2}$$

Calculate \frac{11}{2} to the power of 2 and get \frac{121}{4}.

$$x^{2}=\frac{\left(3\sqrt{3}\right)^{2}}{2^{2}}+\frac{121}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Expand 2^{2}.

$$x^{2}=\frac{\left(3\sqrt{3}\right)^{2}}{4}+\frac{121}{4}$$

Since \frac{\left(3\sqrt{3}\right)^{2}}{4} and \frac{121}{4} have the same denominator, add them by adding their numerators.

$$x^{2}=\frac{\left(3\sqrt{3}\right)^{2}+121}{4}$$

Expand \left(3\sqrt{3}\right)^{2}.

$$x^{2}=\frac{3^{2}\left(\sqrt{3}\right)^{2}+121}{4}$$

Calculate 3 to the power of 2 and get 9.

$$x^{2}=\frac{9\left(\sqrt{3}\right)^{2}+121}{4}$$

The square of \sqrt{3} is 3.

$$x^{2}=\frac{9\times 3+121}{4}$$

Multiply 9 and 3 to get 27.

$$x^{2}=\frac{27+121}{4}$$

Add 27 and 121 to get 148.

$$x^{2}=\frac{148}{4}$$

Divide 148 by 4 to get 37.

$$x^{2}=37$$

Subtract 37 from both sides.

$$x^{2}-37=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -37 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-37\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-37\right)}}{2}$$

Multiply -4 times -37.

$$x=\frac{0±\sqrt{148}}{2}$$

Take the square root of 148.

$$x=\frac{0±2\sqrt{37}}{2}$$

Now solve the equation x=\frac{0±2\sqrt{37}}{2} when ± is plus.

$$x=\sqrt{37}$$

Now solve the equation x=\frac{0±2\sqrt{37}}{2} when ± is minus.

$$x=-\sqrt{37}$$

The equation is now solved.

$$x=\sqrt{37} x=-\sqrt{37}$$