Solve (9x+1)(9x-1)=1

$$( 9 x + 1 ) ( 9 x - 1 ) = 1$$

Solve for x

$$x=\frac{\sqrt{2}}{9}\approx 0.15713484$$

$$x=-\frac{\sqrt{2}}{9}\approx -0.15713484$$

Steps by Finding Square Root

Consider \left(9x+1\right)\left(9x-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$\left(9x\right)^{2}-1=1$$

Expand \left(9x\right)^{2}.

$$9^{2}x^{2}-1=1$$

Calculate 9 to the power of 2 and get 81.

$$81x^{2}-1=1$$

Add 1 to both sides.

$$81x^{2}=1+1$$

Add 1 and 1 to get 2.

$$81x^{2}=2$$

Divide both sides by 81.

$$x^{2}=\frac{2}{81}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{2}}{9} x=-\frac{\sqrt{2}}{9}$$

Steps Using the Quadratic Formula

Consider \left(9x+1\right)\left(9x-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$\left(9x\right)^{2}-1=1$$

Expand \left(9x\right)^{2}.

$$9^{2}x^{2}-1=1$$

Calculate 9 to the power of 2 and get 81.

$$81x^{2}-1=1$$

Subtract 1 from both sides.

$$81x^{2}-1-1=0$$

Subtract 1 from -1 to get -2.

$$81x^{2}-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 81 for a, 0 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 81\left(-2\right)}}{2\times 81}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 81\left(-2\right)}}{2\times 81}$$

Multiply -4 times 81.

$$x=\frac{0±\sqrt{-324\left(-2\right)}}{2\times 81}$$

Multiply -324 times -2.

$$x=\frac{0±\sqrt{648}}{2\times 81}$$

Take the square root of 648.

$$x=\frac{0±18\sqrt{2}}{2\times 81}$$

Multiply 2 times 81.

$$x=\frac{0±18\sqrt{2}}{162}$$

Now solve the equation x=\frac{0±18\sqrt{2}}{162} when ± is plus.

$$x=\frac{\sqrt{2}}{9}$$

Now solve the equation x=\frac{0±18\sqrt{2}}{162} when ± is minus.

$$x=-\frac{\sqrt{2}}{9}$$

The equation is now solved.

$$x=\frac{\sqrt{2}}{9} x=-\frac{\sqrt{2}}{9}$$