Solve -(x+1)(x-3)/(x-1)^2+1=42

$$\frac { - ( x + 1 ) ( x - 3 ) } { ( x - 1 ) ^ { 2 } } + 1 = 42$$

Solve for x

$$x=\frac{\sqrt{42}}{21}+1\approx 1.3086067$$

$$x=-\frac{\sqrt{42}}{21}+1\approx 0.6913933$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)^{2}.

$$\left(-\left(x+1\right)\right)\left(x-3\right)+\left(x-1\right)^{2}=42\left(x-1\right)^{2}$$

To find the opposite of x+1, find the opposite of each term.

$$\left(-x-1\right)\left(x-3\right)+\left(x-1\right)^{2}=42\left(x-1\right)^{2}$$

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

$$-x^{2}+2x+3+\left(x-1\right)^{2}=42\left(x-1\right)^{2}$$

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

$$-x^{2}+2x+3+x^{2}-2x+1=42\left(x-1\right)^{2}$$

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

$$2x+3-2x+1=42\left(x-1\right)^{2}$$

Combine 2x and -2x to get 0.

$$3+1=42\left(x-1\right)^{2}$$

Add 3 and 1 to get 4.

$$4=42\left(x-1\right)^{2}$$

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

$$4=42\left(x^{2}-2x+1\right)$$

Use the distributive property to multiply 42 by x^{2}-2x+1.

$$4=42x^{2}-84x+42$$

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

$$42x^{2}-84x+42=4$$

Subtract 4 from both sides.

$$42x^{2}-84x+42-4=0$$

Subtract 4 from 42 to get 38.

$$42x^{2}-84x+38=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 42 for a, -84 for b, and 38 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-84\right)±\sqrt{\left(-84\right)^{2}-4\times 42\times 38}}{2\times 42}$$

Square -84.

$$x=\frac{-\left(-84\right)±\sqrt{7056-4\times 42\times 38}}{2\times 42}$$

Multiply -4 times 42.

$$x=\frac{-\left(-84\right)±\sqrt{7056-168\times 38}}{2\times 42}$$

Multiply -168 times 38.

$$x=\frac{-\left(-84\right)±\sqrt{7056-6384}}{2\times 42}$$

Add 7056 to -6384.

$$x=\frac{-\left(-84\right)±\sqrt{672}}{2\times 42}$$

Take the square root of 672.

$$x=\frac{-\left(-84\right)±4\sqrt{42}}{2\times 42}$$

The opposite of -84 is 84.

$$x=\frac{84±4\sqrt{42}}{2\times 42}$$

Multiply 2 times 42.

$$x=\frac{84±4\sqrt{42}}{84}$$

Now solve the equation x=\frac{84±4\sqrt{42}}{84} when ± is plus. Add 84 to 4\sqrt{42}.

$$x=\frac{4\sqrt{42}+84}{84}$$

Divide 84+4\sqrt{42} by 84.

$$x=\frac{\sqrt{42}}{21}+1$$

Now solve the equation x=\frac{84±4\sqrt{42}}{84} when ± is minus. Subtract 4\sqrt{42} from 84.

$$x=\frac{84-4\sqrt{42}}{84}$$

Divide 84-4\sqrt{42} by 84.

$$x=-\frac{\sqrt{42}}{21}+1$$

The equation is now solved.

$$x=\frac{\sqrt{42}}{21}+1 x=-\frac{\sqrt{42}}{21}+1$$

Steps for Completing the Square

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)^{2}.

$$\left(-\left(x+1\right)\right)\left(x-3\right)+\left(x-1\right)^{2}=42\left(x-1\right)^{2}$$

To find the opposite of x+1, find the opposite of each term.

$$\left(-x-1\right)\left(x-3\right)+\left(x-1\right)^{2}=42\left(x-1\right)^{2}$$

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

$$-x^{2}+2x+3+\left(x-1\right)^{2}=42\left(x-1\right)^{2}$$

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

$$-x^{2}+2x+3+x^{2}-2x+1=42\left(x-1\right)^{2}$$

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

$$2x+3-2x+1=42\left(x-1\right)^{2}$$

Combine 2x and -2x to get 0.

$$3+1=42\left(x-1\right)^{2}$$

Add 3 and 1 to get 4.

$$4=42\left(x-1\right)^{2}$$

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

$$4=42\left(x^{2}-2x+1\right)$$

Use the distributive property to multiply 42 by x^{2}-2x+1.

$$4=42x^{2}-84x+42$$

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

$$42x^{2}-84x+42=4$$

Subtract 42 from both sides.

$$42x^{2}-84x=4-42$$

Subtract 42 from 4 to get -38.

$$42x^{2}-84x=-38$$

Divide both sides by 42.

$$\frac{42x^{2}-84x}{42}=-\frac{38}{42}$$

Dividing by 42 undoes the multiplication by 42.

$$x^{2}+\left(-\frac{84}{42}\right)x=-\frac{38}{42}$$

Divide -84 by 42.

$$x^{2}-2x=-\frac{38}{42}$$

Reduce the fraction \frac{-38}{42} to lowest terms by extracting and canceling out 2.

$$x^{2}-2x=-\frac{19}{21}$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=-\frac{19}{21}+1$$

Add -\frac{19}{21} to 1.

$$x^{2}-2x+1=\frac{2}{21}$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=\frac{2}{21}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{2}{21}}$$

Simplify.

$$x-1=\frac{\sqrt{42}}{21} x-1=-\frac{\sqrt{42}}{21}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{42}}{21}+1 x=-\frac{\sqrt{42}}{21}+1$$