Solve 5/x-5+x/x-2=-6x+17/(x-5)(x-2)

$$\frac { 5 } { x - 5 } + \frac { x } { x - 2 } = \frac { - 6 x + 17 } { ( x - 5 ) ( x - 2 ) }$$

Solve for x

$$x=-9$$

$$x=3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 2,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x-2\right), the least common multiple of x-5,x-2,\left(x-5\right)\left(x-2\right).

$$\left(x-2\right)\times 5+\left(x-5\right)x=-6x+17$$

Use the distributive property to multiply x-2 by 5.

$$5x-10+\left(x-5\right)x=-6x+17$$

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

$$5x-10+x^{2}-5x=-6x+17$$

Combine 5x and -5x to get 0.

$$-10+x^{2}=-6x+17$$

Add 6x to both sides.

$$-10+x^{2}+6x=17$$

Subtract 17 from both sides.

$$-10+x^{2}+6x-17=0$$

Subtract 17 from -10 to get -27.

$$-27+x^{2}+6x=0$$

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.

$$x^{2}+6x-27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-27\right)}}{2}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-27\right)}}{2}$$

Multiply -4 times -27.

$$x=\frac{-6±\sqrt{36+108}}{2}$$

Add 36 to 108.

$$x=\frac{-6±\sqrt{144}}{2}$$

Take the square root of 144.

$$x=\frac{-6±12}{2}$$

Now solve the equation x=\frac{-6±12}{2} when ± is plus. Add -6 to 12.

$$x=\frac{6}{2}$$

Divide 6 by 2.

$$x=3$$

Now solve the equation x=\frac{-6±12}{2} when ± is minus. Subtract 12 from -6.

$$x=-\frac{18}{2}$$

Divide -18 by 2.

$$x=-9$$

The equation is now solved.

$$x=3 x=-9$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 2,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x-2\right), the least common multiple of x-5,x-2,\left(x-5\right)\left(x-2\right).

$$\left(x-2\right)\times 5+\left(x-5\right)x=-6x+17$$

Use the distributive property to multiply x-2 by 5.

$$5x-10+\left(x-5\right)x=-6x+17$$

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

$$5x-10+x^{2}-5x=-6x+17$$

Combine 5x and -5x to get 0.

$$-10+x^{2}=-6x+17$$

Add 6x to both sides.

$$-10+x^{2}+6x=17$$

Add 10 to both sides.

$$x^{2}+6x=17+10$$

Add 17 and 10 to get 27.

$$x^{2}+6x=27$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=27+3^{2}$$

Square 3.

$$x^{2}+6x+9=27+9$$

Add 27 to 9.

$$x^{2}+6x+9=36$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{36}$$

Simplify.

$$x+3=6 x+3=-6$$

Subtract 3 from both sides of the equation.

$$x=3 x=-9$$