Solve {left(9/41right)}^2+x^2=1

$${ \left( \frac{ 9 }{ 41 } \right) }^{ 2 } + { x }^{ 2 } = 1$$

Solve for x

$$x=\frac{40}{41}\approx 0.975609756$$

$$x=-\frac{40}{41}\approx -0.975609756$$

Steps Using Difference of Squares

Calculate \frac{9}{41} to the power of 2 and get \frac{81}{1681}.

$$\frac{81}{1681}+x^{2}=1$$

Subtract 1 from both sides.

$$\frac{81}{1681}+x^{2}-1=0$$

Subtract 1 from \frac{81}{1681} to get -\frac{1600}{1681}.

$$-\frac{1600}{1681}+x^{2}=0$$

Multiply both sides by 1681.

$$-1600+1681x^{2}=0$$

Consider -1600+1681x^{2}. Rewrite -1600+1681x^{2} as \left(41x\right)^{2}-40^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(41x-40\right)\left(41x+40\right)=0$$

To find equation solutions, solve 41x-40=0 and 41x+40=0.

$$x=\frac{40}{41} x=-\frac{40}{41}$$

Steps by Finding Square Root

Calculate \frac{9}{41} to the power of 2 and get \frac{81}{1681}.

$$\frac{81}{1681}+x^{2}=1$$

Subtract \frac{81}{1681} from both sides.

$$x^{2}=1-\frac{81}{1681}$$

Subtract \frac{81}{1681} from 1 to get \frac{1600}{1681}.

$$x^{2}=\frac{1600}{1681}$$

Take the square root of both sides of the equation.

$$x=\frac{40}{41} x=-\frac{40}{41}$$

Steps Using the Quadratic Formula

Calculate \frac{9}{41} to the power of 2 and get \frac{81}{1681}.

$$\frac{81}{1681}+x^{2}=1$$

Subtract 1 from both sides.

$$\frac{81}{1681}+x^{2}-1=0$$

Subtract 1 from \frac{81}{1681} to get -\frac{1600}{1681}.

$$-\frac{1600}{1681}+x^{2}=0$$

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.

$$x^{2}-\frac{1600}{1681}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -\frac{1600}{1681} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{1600}{1681}\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{1600}{1681}\right)}}{2}$$

Multiply -4 times -\frac{1600}{1681}.

$$x=\frac{0±\sqrt{\frac{6400}{1681}}}{2}$$

Take the square root of \frac{6400}{1681}.

$$x=\frac{0±\frac{80}{41}}{2}$$

Now solve the equation x=\frac{0±\frac{80}{41}}{2} when ± is plus.

$$x=\frac{40}{41}$$

Now solve the equation x=\frac{0±\frac{80}{41}}{2} when ± is minus.

$$x=-\frac{40}{41}$$

The equation is now solved.

$$x=\frac{40}{41} x=-\frac{40}{41}$$