Solve (13x-1)^2-(12x+3)^2/(4x+5)^2-41(x-1)(x+1)=-1

$$\frac { ( 13 x - 1 ) ^ { 2 } - ( 12 x + 3 ) ^ { 2 } } { ( 4 x + 5 ) ^ { 2 } - 41 ( x - 1 ) ( x + 1 ) } = - 1$$

Solve for x

$$x=1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 25\left(x-\left(-\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)\left(x-\left(\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right).

$$-\left(\left(13x-1\right)^{2}-\left(12x+3\right)^{2}\right)=-25\left(x-\left(-\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)\left(x-\left(\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(13x-1\right)^{2}.

$$-\left(169x^{2}-26x+1-\left(12x+3\right)^{2}\right)=-25\left(x-\left(-\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)\left(x-\left(\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(12x+3\right)^{2}.

$$-\left(169x^{2}-26x+1-\left(144x^{2}+72x+9\right)\right)=-25\left(x-\left(-\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)\left(x-\left(\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)$$

To find the opposite of 144x^{2}+72x+9, find the opposite of each term.

$$-\left(169x^{2}-26x+1-144x^{2}-72x-9\right)=-25\left(x-\left(-\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)\left(x-\left(\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)$$

Combine 169x^{2} and -144x^{2} to get 25x^{2}.

$$-\left(25x^{2}-26x+1-72x-9\right)=-25\left(x-\left(-\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)\left(x-\left(\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)$$

Combine -26x and -72x to get -98x.

$$-\left(25x^{2}-98x+1-9\right)=-25\left(x-\left(-\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)\left(x-\left(\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)$$

Subtract 9 from 1 to get -8.

$$-\left(25x^{2}-98x-8\right)=-25\left(x-\left(-\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)\left(x-\left(\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)$$

To find the opposite of 25x^{2}-98x-8, find the opposite of each term.

$$-25x^{2}+98x+8=-25\left(x-\left(-\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)\left(x-\left(\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)$$

To find the opposite of -\frac{1}{5}\sqrt{82}+\frac{4}{5}, find the opposite of each term.

$$-25x^{2}+98x+8=-25\left(x+\frac{1}{5}\sqrt{82}-\frac{4}{5}\right)\left(x-\left(\frac{1}{5}\sqrt{82}+\frac{4}{5}\right)\right)$$

To find the opposite of \frac{1}{5}\sqrt{82}+\frac{4}{5}, find the opposite of each term.

$$-25x^{2}+98x+8=-25\left(x+\frac{1}{5}\sqrt{82}-\frac{4}{5}\right)\left(x-\frac{1}{5}\sqrt{82}-\frac{4}{5}\right)$$

Use the distributive property to multiply -25 by x+\frac{1}{5}\sqrt{82}-\frac{4}{5}.

$$-25x^{2}+98x+8=\left(-25x-5\sqrt{82}+20\right)\left(x-\frac{1}{5}\sqrt{82}-\frac{4}{5}\right)$$

Use the distributive property to multiply -25x-5\sqrt{82}+20 by x-\frac{1}{5}\sqrt{82}-\frac{4}{5} and combine like terms.

$$-25x^{2}+98x+8=-25x^{2}+40x+\left(\sqrt{82}\right)^{2}-16$$

The square of \sqrt{82} is 82.

$$-25x^{2}+98x+8=-25x^{2}+40x+82-16$$

Subtract 16 from 82 to get 66.

$$-25x^{2}+98x+8=-25x^{2}+40x+66$$

Add 25x^{2} to both sides.

$$-25x^{2}+98x+8+25x^{2}=40x+66$$

Combine -25x^{2} and 25x^{2} to get 0.

$$98x+8=40x+66$$

Subtract 40x from both sides.

$$98x+8-40x=66$$

Combine 98x and -40x to get 58x.

$$58x+8=66$$

Subtract 8 from both sides.

$$58x=66-8$$

Subtract 8 from 66 to get 58.

$$58x=58$$

Divide both sides by 58.

$$x=\frac{58}{58}$$

Divide 58 by 58 to get 1.

$$x=1$$