Solve (5/13)^2+x^2=1

$$( \frac { 5 } { 13 } ) ^ { 2 } + x ^ { 2 } = 1$$

Solve for x

$$x=\frac{12}{13}\approx 0.923076923$$

$$x=-\frac{12}{13}\approx -0.923076923$$

Steps Using Difference of Squares

Calculate \frac{5}{13} to the power of 2 and get \frac{25}{169}.

$$\frac{25}{169}+x^{2}=1$$

Subtract 1 from both sides.

$$\frac{25}{169}+x^{2}-1=0$$

Subtract 1 from \frac{25}{169} to get -\frac{144}{169}.

$$-\frac{144}{169}+x^{2}=0$$

Multiply both sides by 169.

$$-144+169x^{2}=0$$

Consider -144+169x^{2}. Rewrite -144+169x^{2} as \left(13x\right)^{2}-12^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(13x-12\right)\left(13x+12\right)=0$$

To find equation solutions, solve 13x-12=0 and 13x+12=0.

$$x=\frac{12}{13} x=-\frac{12}{13}$$

Steps by Finding Square Root

Calculate \frac{5}{13} to the power of 2 and get \frac{25}{169}.

$$\frac{25}{169}+x^{2}=1$$

Subtract \frac{25}{169} from both sides.

$$x^{2}=1-\frac{25}{169}$$

Subtract \frac{25}{169} from 1 to get \frac{144}{169}.

$$x^{2}=\frac{144}{169}$$

Take the square root of both sides of the equation.

$$x=\frac{12}{13} x=-\frac{12}{13}$$

Steps Using the Quadratic Formula

Calculate \frac{5}{13} to the power of 2 and get \frac{25}{169}.

$$\frac{25}{169}+x^{2}=1$$

Subtract 1 from both sides.

$$\frac{25}{169}+x^{2}-1=0$$

Subtract 1 from \frac{25}{169} to get -\frac{144}{169}.

$$-\frac{144}{169}+x^{2}=0$$

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.

$$x^{2}-\frac{144}{169}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -\frac{144}{169} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{144}{169}\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{144}{169}\right)}}{2}$$

Multiply -4 times -\frac{144}{169}.

$$x=\frac{0±\sqrt{\frac{576}{169}}}{2}$$

Take the square root of \frac{576}{169}.

$$x=\frac{0±\frac{24}{13}}{2}$$

Now solve the equation x=\frac{0±\frac{24}{13}}{2} when ± is plus.

$$x=\frac{12}{13}$$

Now solve the equation x=\frac{0±\frac{24}{13}}{2} when ± is minus.

$$x=-\frac{12}{13}$$

The equation is now solved.

$$x=\frac{12}{13} x=-\frac{12}{13}$$