Solve x^2+(frac{3sqrt{3}}{14})^2=1

$$x ^ { 2 } + ( \frac { 3 \sqrt { 3 } } { 14 } ) ^ { 2 } = 1$$

Solve for x

$$x=\frac{13}{14}\approx 0.928571429$$

$$x=-\frac{13}{14}\approx -0.928571429$$

Steps Using Difference of Squares

To raise \frac{3\sqrt{3}}{14} to a power, raise both numerator and denominator to the power and then divide.

$$x^{2}+\frac{\left(3\sqrt{3}\right)^{2}}{14^{2}}=1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2} times \frac{14^{2}}{14^{2}}.

$$\frac{x^{2}\times 14^{2}}{14^{2}}+\frac{\left(3\sqrt{3}\right)^{2}}{14^{2}}=1$$

Since \frac{x^{2}\times 14^{2}}{14^{2}} and \frac{\left(3\sqrt{3}\right)^{2}}{14^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}\times 14^{2}+\left(3\sqrt{3}\right)^{2}}{14^{2}}=1$$

Calculate 14 to the power of 2 and get 196.

$$\frac{x^{2}\times 196+\left(3\sqrt{3}\right)^{2}}{14^{2}}=1$$

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

$$\frac{x^{2}\times 196+3^{2}\left(\sqrt{3}\right)^{2}}{14^{2}}=1$$

Calculate 3 to the power of 2 and get 9.

$$\frac{x^{2}\times 196+9\left(\sqrt{3}\right)^{2}}{14^{2}}=1$$

The square of \sqrt{3} is 3.

$$\frac{x^{2}\times 196+9\times 3}{14^{2}}=1$$

Multiply 9 and 3 to get 27.

$$\frac{x^{2}\times 196+27}{14^{2}}=1$$

Calculate 14 to the power of 2 and get 196.

$$\frac{x^{2}\times 196+27}{196}=1$$

Divide each term of x^{2}\times 196+27 by 196 to get x^{2}+\frac{27}{196}.

$$x^{2}+\frac{27}{196}=1$$

Subtract 1 from both sides.

$$x^{2}+\frac{27}{196}-1=0$$

Subtract 1 from \frac{27}{196} to get -\frac{169}{196}.

$$x^{2}-\frac{169}{196}=0$$

Multiply both sides by 196.

$$196x^{2}-169=0$$

Consider 196x^{2}-169. Rewrite 196x^{2}-169 as \left(14x\right)^{2}-13^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(14x-13\right)\left(14x+13\right)=0$$

To find equation solutions, solve 14x-13=0 and 14x+13=0.

$$x=\frac{13}{14} x=-\frac{13}{14}$$

Steps by Finding Square Root

To raise \frac{3\sqrt{3}}{14} to a power, raise both numerator and denominator to the power and then divide.

$$x^{2}+\frac{\left(3\sqrt{3}\right)^{2}}{14^{2}}=1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2} times \frac{14^{2}}{14^{2}}.

$$\frac{x^{2}\times 14^{2}}{14^{2}}+\frac{\left(3\sqrt{3}\right)^{2}}{14^{2}}=1$$

Since \frac{x^{2}\times 14^{2}}{14^{2}} and \frac{\left(3\sqrt{3}\right)^{2}}{14^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}\times 14^{2}+\left(3\sqrt{3}\right)^{2}}{14^{2}}=1$$

Calculate 14 to the power of 2 and get 196.

$$\frac{x^{2}\times 196+\left(3\sqrt{3}\right)^{2}}{14^{2}}=1$$

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

$$\frac{x^{2}\times 196+3^{2}\left(\sqrt{3}\right)^{2}}{14^{2}}=1$$

Calculate 3 to the power of 2 and get 9.

$$\frac{x^{2}\times 196+9\left(\sqrt{3}\right)^{2}}{14^{2}}=1$$

The square of \sqrt{3} is 3.

$$\frac{x^{2}\times 196+9\times 3}{14^{2}}=1$$

Multiply 9 and 3 to get 27.

$$\frac{x^{2}\times 196+27}{14^{2}}=1$$

Calculate 14 to the power of 2 and get 196.

$$\frac{x^{2}\times 196+27}{196}=1$$

Divide each term of x^{2}\times 196+27 by 196 to get x^{2}+\frac{27}{196}.

$$x^{2}+\frac{27}{196}=1$$

Subtract \frac{27}{196} from both sides.

$$x^{2}=1-\frac{27}{196}$$

Subtract \frac{27}{196} from 1 to get \frac{169}{196}.

$$x^{2}=\frac{169}{196}$$

Take the square root of both sides of the equation.

$$x=\frac{13}{14} x=-\frac{13}{14}$$

Steps Using the Quadratic Formula

To raise \frac{3\sqrt{3}}{14} to a power, raise both numerator and denominator to the power and then divide.

$$x^{2}+\frac{\left(3\sqrt{3}\right)^{2}}{14^{2}}=1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2} times \frac{14^{2}}{14^{2}}.

$$\frac{x^{2}\times 14^{2}}{14^{2}}+\frac{\left(3\sqrt{3}\right)^{2}}{14^{2}}=1$$

Since \frac{x^{2}\times 14^{2}}{14^{2}} and \frac{\left(3\sqrt{3}\right)^{2}}{14^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}\times 14^{2}+\left(3\sqrt{3}\right)^{2}}{14^{2}}=1$$

Calculate 14 to the power of 2 and get 196.

$$\frac{x^{2}\times 196+\left(3\sqrt{3}\right)^{2}}{14^{2}}=1$$

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

$$\frac{x^{2}\times 196+3^{2}\left(\sqrt{3}\right)^{2}}{14^{2}}=1$$

Calculate 3 to the power of 2 and get 9.

$$\frac{x^{2}\times 196+9\left(\sqrt{3}\right)^{2}}{14^{2}}=1$$

The square of \sqrt{3} is 3.

$$\frac{x^{2}\times 196+9\times 3}{14^{2}}=1$$

Multiply 9 and 3 to get 27.

$$\frac{x^{2}\times 196+27}{14^{2}}=1$$

Calculate 14 to the power of 2 and get 196.

$$\frac{x^{2}\times 196+27}{196}=1$$

Divide each term of x^{2}\times 196+27 by 196 to get x^{2}+\frac{27}{196}.

$$x^{2}+\frac{27}{196}=1$$

Subtract 1 from both sides.

$$x^{2}+\frac{27}{196}-1=0$$

Subtract 1 from \frac{27}{196} to get -\frac{169}{196}.

$$x^{2}-\frac{169}{196}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -\frac{169}{196} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{169}{196}\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{169}{196}\right)}}{2}$$

Multiply -4 times -\frac{169}{196}.

$$x=\frac{0±\sqrt{\frac{169}{49}}}{2}$$

Take the square root of \frac{169}{49}.

$$x=\frac{0±\frac{13}{7}}{2}$$

Now solve the equation x=\frac{0±\frac{13}{7}}{2} when ± is plus.

$$x=\frac{13}{14}$$

Now solve the equation x=\frac{0±\frac{13}{7}}{2} when ± is minus.

$$x=-\frac{13}{14}$$

The equation is now solved.

$$x=\frac{13}{14} x=-\frac{13}{14}$$