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Solving Rational Equations Quick Check

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Question
What is the least common denominator of the equation 5x+5−1x2+2x−15=4x2+2x−15?(1 point)
Responses

(x+5)(x2+2x−15)
left parenthesis x plus 5 right parenthesis left parenthesis x squared plus 2 x minus 15 right parenthesis

(x+5)(x2+2x−15)(x2+2x−15)
left parenthesis x plus 5 right parenthesis left parenthesis x squared plus 2 x minus 15 right parenthesis left parenthesis x squared plus 2 x minus 15 right parenthesis

(x+5)
left parenthesis x plus 5 right parenthesis

(x+5)(x−3)

1 answer

To find the least common denominator (LCD) of the given rational equation, we need to first factor all the denominators present in the equation.

  1. Identify the denominators in the equation:

    • On the left side, there is a term \( -1/(x^2 + 2x - 15) \).
    • On the right side, there is a term \( 4/(x^2 + 2x - 15) \).

  2. Factor the polynomial \( x^2 + 2x - 15 \): \[ x^2 + 2x - 15 = (x + 5)(x - 3) \]

  3. Identify all the factors:

    • The left side has a denominator of \( (x + 5)(x - 3) \).
    • The right side also has the same denominator.

Since both sides of the equation share the same denominator \( (x + 5)(x - 3) \), the least common denominator for the equation will be:

\[ \text{LCD} = (x + 5)(x - 3) \]

None of the options provided directly match this, but based on the options:

  • Correct Option: \((x+5)(x−3)\)

So, the answer is:

(x + 5)(x - 3)