Solve x-1=11-x/x-5

$$x - 1 = \frac { 11 - x } { x - 5 }$$

Solve for x

$$x=-1$$

$$x=6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by x-5.

$$\left(x-5\right)x+\left(x-5\right)\left(-1\right)=11-x$$

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

$$x^{2}-5x+\left(x-5\right)\left(-1\right)=11-x$$

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

$$x^{2}-5x-x+5=11-x$$

Combine -5x and -x to get -6x.

$$x^{2}-6x+5=11-x$$

Subtract 11 from both sides.

$$x^{2}-6x+5-11=-x$$

Subtract 11 from 5 to get -6.

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

Add x to both sides.

$$x^{2}-6x-6+x=0$$

Combine -6x and x to get -5x.

$$x^{2}-5x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-6\right)}}{2}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-6\right)}}{2}$$

Multiply -4 times -6.

$$x=\frac{-\left(-5\right)±\sqrt{25+24}}{2}$$

Add 25 to 24.

$$x=\frac{-\left(-5\right)±\sqrt{49}}{2}$$

Take the square root of 49.

$$x=\frac{-\left(-5\right)±7}{2}$$

The opposite of -5 is 5.

$$x=\frac{5±7}{2}$$

Now solve the equation x=\frac{5±7}{2} when ± is plus. Add 5 to 7.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

Now solve the equation x=\frac{5±7}{2} when ± is minus. Subtract 7 from 5.

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

Divide -2 by 2.

$$x=-1$$

The equation is now solved.

$$x=6 x=-1$$

Steps for Completing the Square

Variable x cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by x-5.

$$\left(x-5\right)x+\left(x-5\right)\left(-1\right)=11-x$$

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

$$x^{2}-5x+\left(x-5\right)\left(-1\right)=11-x$$

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

$$x^{2}-5x-x+5=11-x$$

Combine -5x and -x to get -6x.

$$x^{2}-6x+5=11-x$$

Add x to both sides.

$$x^{2}-6x+5+x=11$$

Combine -6x and x to get -5x.

$$x^{2}-5x+5=11$$

Subtract 5 from both sides.

$$x^{2}-5x=11-5$$

Subtract 5 from 11 to get 6.

$$x^{2}-5x=6$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=6+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=6+\frac{25}{4}$$

Add 6 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{49}{4}$$

Factor x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{7}{2} x-\frac{5}{2}=-\frac{7}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=6 x=-1$$