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# Find

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Find all values of x such that:  x/(x - 5) = 4/(x - 4) + 12/(x^2 - 9x + 20)

Dec 31, 2022

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We have:

$$\frac{x}{x-5}=\frac{4}{x-4}+\frac{12}{x^2-9x+20}$$

The best way to approach this is to get rid of the fractions.

We notice that x2-9x+20 = (x-4)(x-5)

Now we multiply both sides by x2-9x+20

$$(x^2-9x+20)(\frac{x}{x-5})=(x^2-9x+20)(\frac{4}{x-4}+\frac{12}{x^2-9x+20})$$

This equals:

$$x(x-4)=4(x-5)+12$$

$$x^2-4x=4x-20+12$$

$$x^2-8x+8=0$$

We notice this is not factorable, so we use the quadratic formula:  (+- means plus minus)

$${x}=\frac{8+-\sqrt{32}}{2}$$

$${x}_{1,2}=4+2\sqrt{2}, 4-2\sqrt{2}$$

Dec 31, 2022