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Question Simplify the expression 5xx+3÷x2−8x+15x2−9\frac{5 x}{x+3} \div \frac{x^{2}-8 x+15}{x^{2}-9}.

Studdy Solution

STEP 1

Assumptions
1. We are given a division of two rational expressions.
2. We need to simplify the expression to its simplest form.
3. We assume that the variable xx is such that the denominators are non-zero (i.e., x≠−3x \neq -3 and x≠±3x \neq \pm3).

STEP 2

Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we can rewrite the division as a multiplication:
5xx+3÷x2−8x+15x2−9=5xx+3×x2−9x2−8x+15\frac{5x}{x+3} \div \frac{x^{2}-8x+15}{x^{2}-9} = \frac{5x}{x+3} \times \frac{x^{2}-9}{x^{2}-8x+15}

STEP 3

Before multiplying, we should factor the quadratic expressions in the numerators and denominators to see if there are common factors that can be cancelled out.

STEP 4

Factor the quadratic expression x2−8x+15x^{2}-8x+15 in the denominator of the second fraction.
The factors of 15 that add up to -8 are -5 and -3. Therefore, we can factor the quadratic as:
x2−8x+15=(x−5)(x−3)x^{2}-8x+15 = (x-5)(x-3)

STEP 5

Factor the quadratic expression x2−9x^{2}-9 in the numerator of the second fraction.
This is a difference of squares, which can be factored as:
x2−9=(x+3)(x−3)x^{2}-9 = (x+3)(x-3)

STEP 6

Now, we rewrite the original expression with the factored forms:
5xx+3×(x+3)(x−3)(x−5)(x−3)\frac{5x}{x+3} \times \frac{(x+3)(x-3)}{(x-5)(x-3)}

STEP 7

Notice that (x+3)(x+3) and (x−3)(x-3) appear in both the numerator and the denominator. We can cancel these common factors, keeping in mind that we are not cancelling across the division, but rather within the multiplication of two fractions.

STEP 8

Cancel the common factors (x+3)(x+3) and (x−3)(x-3):
5xx+3×x+3(x−3)(x−5)x−3=5x1×1x−5\frac{5x}{\cancel{x+3}} \times \frac{\cancel{x+3}(x-3)}{(x-5)\cancel{x-3}} = \frac{5x}{1} \times \frac{1}{x-5}

STEP 9

Now, multiply the remaining expressions:
5x1×1x−5=5xx−5\frac{5x}{1} \times \frac{1}{x-5} = \frac{5x}{x-5}

STEP 10

The expression 5xx−5\frac{5x}{x-5} is already in its simplest form, as there are no common factors left to cancel.
Therefore, the simplified form of the original expression is:
5xx−5\frac{5x}{x-5}

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