Math

Question Simplify the expression x2+5x36x28x+16\frac{x^{2}+5 x-36}{x^{2}-8 x+16} and determine which of the following statements are true: 1) It could represent the difference of xx4\frac{x}{x-4} and 9x4\frac{-9}{x-4}. 2) It could represent the product of x+9(x4)2\frac{x+9}{(x-4)^{2}} and (x4)(x-4). 3) It could represent the sum of a2+5ax28x\frac{a^{2}+5 a}{x^{2}-8 x} and 3616\frac{-36}{16}. 4) It could represent the quotient of (x9)(x-9) and (x4)(x-4). 5) It is equivalent to the rational expression x+9x4\frac{x+9}{x-4}.

Studdy Solution

STEP 1

Assumptions
1. We are given the expression x2+5x36x28x+16\frac{x^{2}+5x-36}{x^{2}-8x+16}.
2. We need to determine the truth of several statements regarding this expression.
3. We will need to factor the numerator and denominator where possible.
4. We will compare the factored form of the given expression to each of the provided statements.

STEP 2

First, let's attempt to factor the numerator x2+5x36x^{2}+5x-36.

STEP 3

We look for two numbers that multiply to 36-36 and add up to 55. These numbers are 99 and 4-4.

STEP 4

We can now factor the numerator:
x2+5x36=(x+9)(x4)x^{2}+5x-36 = (x+9)(x-4)

STEP 5

Next, we factor the denominator x28x+16x^{2}-8x+16.

STEP 6

We look for two numbers that multiply to 1616 and add up to 8-8. These numbers are 4-4 and 4-4.

STEP 7

We can now factor the denominator:
x28x+16=(x4)(x4)=(x4)2x^{2}-8x+16 = (x-4)(x-4) = (x-4)^{2}

STEP 8

Now we rewrite the original expression with the factored numerator and denominator:
x2+5x36x28x+16=(x+9)(x4)(x4)2\frac{x^{2}+5x-36}{x^{2}-8x+16} = \frac{(x+9)(x-4)}{(x-4)^{2}}

STEP 9

We simplify the expression by canceling out one (x4)(x-4) term from the numerator and denominator:
(x+9)(x4)(x4)2=x+9x4\frac{(x+9)(x-4)}{(x-4)^{2}} = \frac{x+9}{x-4}

STEP 10

Now we will evaluate each statement given in the problem:
1. The expression could represent the difference of xx4\frac{x}{x-4} and 9x4\frac{-9}{x-4}.
2. The expression could represent the product of x+9(x4)2\frac{x+9}{(x-4)^{2}} and (x4)(x-4).
3. The expression could represent the sum of a2+5ax28x\frac{a^{2}+5a}{x^{2}-8x} and 3616\frac{-36}{16}.
4. The expression could represent the quotient of (x9)(x-9) and (x4)(x-4).
5. The expression is equivalent to the rational expression x+9x4\frac{x+9}{x-4}.

STEP 11

Let's consider statement 1: The expression could represent the difference of xx4\frac{x}{x-4} and 9x4\frac{-9}{x-4}.

STEP 12

We rewrite the difference of the two fractions:
xx49x4=x+9x4\frac{x}{x-4} - \frac{-9}{x-4} = \frac{x+9}{x-4}

STEP 13

We compare this to the simplified form of the original expression:
x+9x4=x+9x4\frac{x+9}{x-4} = \frac{x+9}{x-4}

STEP 14

Since they are equal, statement 1 is true.

STEP 15

Let's consider statement 2: The expression could represent the product of x+9(x4)2\frac{x+9}{(x-4)^{2}} and (x4)(x-4).

STEP 16

We calculate the product of the two expressions:
x+9(x4)2(x4)=x+9x4\frac{x+9}{(x-4)^{2}} \cdot (x-4) = \frac{x+9}{x-4}

STEP 17

We compare this to the simplified form of the original expression:
x+9x4=x+9x4\frac{x+9}{x-4} = \frac{x+9}{x-4}

STEP 18

Since they are equal, statement 2 is true.

STEP 19

Let's consider statement 3: The expression could represent the sum of a2+5ax28x\frac{a^{2}+5a}{x^{2}-8x} and 3616\frac{-36}{16}.

STEP 20

Since the variable in the numerator of the first fraction is aa and not xx, we cannot directly compare it to the original expression. Moreover, 3616\frac{-36}{16} simplifies to 94-\frac{9}{4}, which cannot be added to a fraction with xx in the denominator without further information. Therefore, statement 3 is false.

STEP 21

Let's consider statement 4: The expression could represent the quotient of (x9)(x-9) and (x4)(x-4).

STEP 22

We write the quotient of the two expressions:
x9x4\frac{x-9}{x-4}

STEP 23

We compare this to the simplified form of the original expression:
x9x4x+9x4\frac{x-9}{x-4} \neq \frac{x+9}{x-4}

STEP 24

Since they are not equal, statement 4 is false.

STEP 25

Let's consider statement 5: The expression is equivalent to the rational expression x+9x4\frac{x+9}{x-4}.

STEP 26

We compare this to the simplified form of the original expression:
x+9x4=x+9x4\frac{x+9}{x-4} = \frac{x+9}{x-4}

STEP 27

Since they are equal, statement 5 is true.

STEP 28

To summarize, the true statements are:
1. The expression could represent the difference of xx4\frac{x}{x-4} and 9x4\frac{-9}{x-4}.
2. The expression could represent the product of x+9(x4)2\frac{x+9}{(x-4)^{2}} and (x4)(x-4).
5. The expression is equivalent to the rational expression x+9x4\frac{x+9}{x-4}.

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