Solve 1/x^2-8x+7=x+7/x-1

$$\frac { 1 } { x ^ { 2 } - 8 x + 7 } = \frac { x + 7 } { x - 1 }$$

Solve for x

$$x=5\sqrt{2}\approx 7.071067812$$

$$x=-5\sqrt{2}\approx -7.071067812$$

Steps by Finding Square Root

Variable x cannot be equal to any of the values 1,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x-1\right), the least common multiple of x^{2}-8x+7,x-1.

$$1=\left(x-7\right)\left(x+7\right)$$

Consider \left(x-7\right)\left(x+7\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 7.

$$1=x^{2}-49$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-49=1$$

Add 49 to both sides.

$$x^{2}=1+49$$

Add 1 and 49 to get 50.

$$x^{2}=50$$

Take the square root of both sides of the equation.

$$x=5\sqrt{2} x=-5\sqrt{2}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 1,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x-1\right), the least common multiple of x^{2}-8x+7,x-1.

$$1=\left(x-7\right)\left(x+7\right)$$

Consider \left(x-7\right)\left(x+7\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 7.

$$1=x^{2}-49$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-49=1$$

Subtract 1 from both sides.

$$x^{2}-49-1=0$$

Subtract 1 from -49 to get -50.

$$x^{2}-50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-50\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-50\right)}}{2}$$

Multiply -4 times -50.

$$x=\frac{0±\sqrt{200}}{2}$$

Take the square root of 200.

$$x=\frac{0±10\sqrt{2}}{2}$$

Now solve the equation x=\frac{0±10\sqrt{2}}{2} when ± is plus.

$$x=5\sqrt{2}$$

Now solve the equation x=\frac{0±10\sqrt{2}}{2} when ± is minus.

$$x=-5\sqrt{2}$$

The equation is now solved.

$$x=5\sqrt{2} x=-5\sqrt{2}$$