Solve (x-1)(x+1)=5(1-3x)(1+3x)+1

$$(x-1)(x+1)=5(1-3x)(1+3x)+1$$

Solve for x

$$x=\frac{\sqrt{322}}{46}\approx 0.390094749$$

$$x=-\frac{\sqrt{322}}{46}\approx -0.390094749$$

Steps by Finding Square Root

Consider \left(x-1\right)\left(x+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.

$$x^{2}-1=5\left(1-3x\right)\left(1+3x\right)+1$$

Use the distributive property to multiply 5 by 1-3x.

$$x^{2}-1=\left(5-15x\right)\left(1+3x\right)+1$$

Use the distributive property to multiply 5-15x by 1+3x and combine like terms.

$$x^{2}-1=5-45x^{2}+1$$

Add 5 and 1 to get 6.

$$x^{2}-1=6-45x^{2}$$

Add 45x^{2} to both sides.

$$x^{2}-1+45x^{2}=6$$

Combine x^{2} and 45x^{2} to get 46x^{2}.

$$46x^{2}-1=6$$

Add 1 to both sides.

$$46x^{2}=6+1$$

Add 6 and 1 to get 7.

$$46x^{2}=7$$

Divide both sides by 46.

$$x^{2}=\frac{7}{46}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{322}}{46} x=-\frac{\sqrt{322}}{46}$$

Steps Using the Quadratic Formula

Consider \left(x-1\right)\left(x+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.

$$x^{2}-1=5\left(1-3x\right)\left(1+3x\right)+1$$

Use the distributive property to multiply 5 by 1-3x.

$$x^{2}-1=\left(5-15x\right)\left(1+3x\right)+1$$

Use the distributive property to multiply 5-15x by 1+3x and combine like terms.

$$x^{2}-1=5-45x^{2}+1$$

Add 5 and 1 to get 6.

$$x^{2}-1=6-45x^{2}$$

Subtract 6 from both sides.

$$x^{2}-1-6=-45x^{2}$$

Subtract 6 from -1 to get -7.

$$x^{2}-7=-45x^{2}$$

Add 45x^{2} to both sides.

$$x^{2}-7+45x^{2}=0$$

Combine x^{2} and 45x^{2} to get 46x^{2}.

$$46x^{2}-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 46 for a, 0 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 46\left(-7\right)}}{2\times 46}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 46\left(-7\right)}}{2\times 46}$$

Multiply -4 times 46.

$$x=\frac{0±\sqrt{-184\left(-7\right)}}{2\times 46}$$

Multiply -184 times -7.

$$x=\frac{0±\sqrt{1288}}{2\times 46}$$

Take the square root of 1288.

$$x=\frac{0±2\sqrt{322}}{2\times 46}$$

Multiply 2 times 46.

$$x=\frac{0±2\sqrt{322}}{92}$$

Now solve the equation x=\frac{0±2\sqrt{322}}{92} when ± is plus.

$$x=\frac{\sqrt{322}}{46}$$

Now solve the equation x=\frac{0±2\sqrt{322}}{92} when ± is minus.

$$x=-\frac{\sqrt{322}}{46}$$

The equation is now solved.

$$x=\frac{\sqrt{322}}{46} x=-\frac{\sqrt{322}}{46}$$