Solve x-1+(4div(x-1))=5

$$x-1+(4 \div (x-1))=5$$

Solve for x

$$x=5$$

$$x=2$$

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 x-1.

$$\left(x-1\right)x+\left(x-1\right)\left(-1\right)+4=5\left(x-1\right)$$

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

$$x^{2}-x+\left(x-1\right)\left(-1\right)+4=5\left(x-1\right)$$

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

$$x^{2}-x-x+1+4=5\left(x-1\right)$$

Combine -x and -x to get -2x.

$$x^{2}-2x+1+4=5\left(x-1\right)$$

Add 1 and 4 to get 5.

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

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

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

Subtract 5x from both sides.

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

Combine -2x and -5x to get -7x.

$$x^{2}-7x+5=-5$$

Add 5 to both sides.

$$x^{2}-7x+5+5=0$$

Add 5 and 5 to get 10.

$$x^{2}-7x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -7 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 10}}{2}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 10}}{2}$$

Multiply -4 times 10.

$$x=\frac{-\left(-7\right)±\sqrt{49-40}}{2}$$

Add 49 to -40.

$$x=\frac{-\left(-7\right)±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-\left(-7\right)±3}{2}$$

The opposite of -7 is 7.

$$x=\frac{7±3}{2}$$

Now solve the equation x=\frac{7±3}{2} when ± is plus. Add 7 to 3.

$$x=\frac{10}{2}$$

Divide 10 by 2.

$$x=5$$

Now solve the equation x=\frac{7±3}{2} when ± is minus. Subtract 3 from 7.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

The equation is now solved.

$$x=5 x=2$$

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 x-1.

$$\left(x-1\right)x+\left(x-1\right)\left(-1\right)+4=5\left(x-1\right)$$

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

$$x^{2}-x+\left(x-1\right)\left(-1\right)+4=5\left(x-1\right)$$

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

$$x^{2}-x-x+1+4=5\left(x-1\right)$$

Combine -x and -x to get -2x.

$$x^{2}-2x+1+4=5\left(x-1\right)$$

Add 1 and 4 to get 5.

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

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

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

Subtract 5x from both sides.

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

Combine -2x and -5x to get -7x.

$$x^{2}-7x+5=-5$$

Subtract 5 from both sides.

$$x^{2}-7x=-5-5$$

Subtract 5 from -5 to get -10.

$$x^{2}-7x=-10$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-10+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=-10+\frac{49}{4}$$

Add -10 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{9}{4}$$

Factor x^{2}-7x+\frac{49}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{3}{2} x-\frac{7}{2}=-\frac{3}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=5 x=2$$