Solving Rational Equations Quick Check

Question

3 of 53 of 5 Items
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)

Answer 1

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)