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Elden [556K]
3 years ago
15

For the rational expression 2x^2-7x-15/x^2-25, do both of the following,

Mathematics
1 answer:
Mrrafil [7]3 years ago
3 0

Answer:

a. \frac{2x + 3}{x+5}

b. x ≠ -5 (Vertical asymptote) and x ≠ 5 (Hole)

Step-by-step explanation:

Factor the numerator (Grouping):

2x^2 - 7x - 15              

Two numbers that multiply to -30 and add to -7 = -3 and 10

[2x^2 - 10x] + [3x - 15]

(2x + 3)(x - 5)

Factor the denominator (Difference of Two Squares):

x^2 - 25 = (x +5)(x-5)

Factored Expression:

(x - 5) can be factored out of top and bottom as a hole-

\frac{(2x + 3)(x - 5)}{(x + 5)(x - 5)} = \frac{2x + 3}{x+5}

Variable Restrictions:

Denominator ≠ 0

x + 5 = 0\\x = -5

Vertical asymptote at x = -5 ⇒ x ≠ -5

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<h2>Steps:</h2>

(Let x = greater number and y = lesser number)

So this question is asking us for a system of equations. Using the info they provide, we can form these two equations:

x=y+7\ \textsf{("The greater of the numbers is 7 more than the lesser.")}\\3x=4y+5\ \textsf{("Three times the greater number is 5 more than 4 times the lesser number.")}

So for this, we will be using the substitution method. Since we know that x = y + 7, substitute x with (y + 7) in the second equation as such:

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From here we can solve for y. Firstly, distribute 3 so that it multiplies with y and 7:

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Next, subtract 3y on both sides of the equation:

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Lastly, subtract 5 on both sides of the equation:

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Now that we know the value of y, we can substitute it into either equation to solve for x:

x=16+7\\x=23\\\\3x=4(16)+5\\3x=64+5\\3x=69\\x=23

<h2>Answer:</h2>

<u>In short, 16 is the lesser number and 23 is the greater number.</u>

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I hope my answer has come to your help. God bless and have a nice day ahead!
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