What two rational expressions sum to 3x−4/x2+x−12?

Enter your answer by filling in the boxes. Enter your answer so that each rational expression is in simplified form

Attachment image

2 Answers

Relevance
  • 1 month ago
    Favorite Answer

    (3x - 4) / (x^2 + x - 12) = (3x - 4) / [(x + 4)(x - 3)]

    (3x - 4) / [(x + 4)(x - 3)] = a / (x + 4) + b / (x - 3)

    Multiply through by (x + 4) (x - 3) to give:

    (3x - 4) = a(x - 3) + b(x + 4)

    3x - 4 = ax - 3a + bx + 4b

    3x - 4 = x(a + b) + (-3a + 4b)

    This means that:

    3 = a + b, and -4 = -3a + 4b

    or 

    9 = 3a + 3b

    -4 = -3a + 4b

    Adding gives:

    5 = 7b

    b = 5/7

    a = 3 - (5/7) = 21/7 - 5/7 = 16/7

    (3x - 4) / (x^2 + x - 12) = (16/7) / (x + 4) + (5/7) / (x - 3) 

  • 1 month ago

    a/(x + 4) + b/(x - 3) = (3x - 4)/(x^2 + x - 12) for some a and b.

    (x - 3)/(x - 3) * a/(x + 4) + (x + 4)/(x + 4) * b/(x - 3) = (3x - 4)/(x^2 + x - 12)

    a(x - 3) + b(x + 4) = 3x - 4

    ax + bx - 3a + 4b = 3x - 4

    solve this system of equations for a and b:

    a + b = 3

    -3a + 4b = -4

    a = 3 - b

    -3(3 - b) + 4b = -4

    -9 + 3b + 4b = -4

    7b = 5

    b = 5/7

    a = 3 - 5/7 = 16/7

    16/(7(x + 4)) + 5/(7(x - 3)) = (3x - 4)/(x^2 + x - 12)

Still have questions? Get your answers by asking now.