Solve 2x-11/x-6=x-4

$$\frac { 2 x - 11 } { x - 6 } = x - 4$$

Solve for x

$$x=5$$

$$x=7$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by x-6.

$$2x-11=\left(x-6\right)x+\left(x-6\right)\left(-4\right)$$

Use the distributive property to multiply x-6 by x.

$$2x-11=x^{2}-6x+\left(x-6\right)\left(-4\right)$$

Use the distributive property to multiply x-6 by -4.

$$2x-11=x^{2}-6x-4x+24$$

Combine -6x and -4x to get -10x.

$$2x-11=x^{2}-10x+24$$

Subtract x^{2} from both sides.

$$2x-11-x^{2}=-10x+24$$

Add 10x to both sides.

$$2x-11-x^{2}+10x=24$$

Combine 2x and 10x to get 12x.

$$12x-11-x^{2}=24$$

Subtract 24 from both sides.

$$12x-11-x^{2}-24=0$$

Subtract 24 from -11 to get -35.

$$12x-35-x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+12x-35=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 12 for b, and -35 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-1\right)\left(-35\right)}}{2\left(-1\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-1\right)\left(-35\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-12±\sqrt{144+4\left(-35\right)}}{2\left(-1\right)}$$

Multiply 4 times -35.

$$x=\frac{-12±\sqrt{144-140}}{2\left(-1\right)}$$

Add 144 to -140.

$$x=\frac{-12±\sqrt{4}}{2\left(-1\right)}$$

Take the square root of 4.

$$x=\frac{-12±2}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-12±2}{-2}$$

Now solve the equation x=\frac{-12±2}{-2} when ± is plus. Add -12 to 2.

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

Divide -10 by -2.

$$x=5$$

Now solve the equation x=\frac{-12±2}{-2} when ± is minus. Subtract 2 from -12.

$$x=-\frac{14}{-2}$$

Divide -14 by -2.

$$x=7$$

The equation is now solved.

$$x=5 x=7$$

Steps for Completing the Square

Variable x cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by x-6.

$$2x-11=\left(x-6\right)x+\left(x-6\right)\left(-4\right)$$

Use the distributive property to multiply x-6 by x.

$$2x-11=x^{2}-6x+\left(x-6\right)\left(-4\right)$$

Use the distributive property to multiply x-6 by -4.

$$2x-11=x^{2}-6x-4x+24$$

Combine -6x and -4x to get -10x.

$$2x-11=x^{2}-10x+24$$

Subtract x^{2} from both sides.

$$2x-11-x^{2}=-10x+24$$

Add 10x to both sides.

$$2x-11-x^{2}+10x=24$$

Combine 2x and 10x to get 12x.

$$12x-11-x^{2}=24$$

Add 11 to both sides.

$$12x-x^{2}=24+11$$

Add 24 and 11 to get 35.

$$12x-x^{2}=35$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}+12x=35$$

Divide both sides by -1.

$$\frac{-x^{2}+12x}{-1}=\frac{35}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{12}{-1}x=\frac{35}{-1}$$

Divide 12 by -1.

$$x^{2}-12x=\frac{35}{-1}$$

Divide 35 by -1.

$$x^{2}-12x=-35$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-12x+\left(-6\right)^{2}=-35+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=-35+36$$

Add -35 to 36.

$$x^{2}-12x+36=1$$

Factor x^{2}-12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-6\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{1}$$

Simplify.

$$x-6=1 x-6=-1$$

Add 6 to both sides of the equation.

$$x=7 x=5$$