Solutions

  1. k = 6

    If −4 is a solution, then you know that (x + 4) must be one of the factors of the quadratic equation. The other factor is (x + ?). You know that the product of 4 and ? must be equal to 8; thus, the other factor is (x + 2). You know that the sum of 4 and 2 must be equal to k. Therefore, k = 6.

  2. (A)

    If the solutions to the equation are 8 and −4, the factored form of the equation is:

    (x − 8)(x + 4) = 0

    Distributed, this equals: x2 − 4x − 32 = 0.

  3. y = {−4, −6}

    Simplify and factor to solve.

    16 − y2 = 10(4 + y)
    16 − y2 = 40 + 10y
    y2 + 10y + 24 = 0
    (y + 4)(y + 6) = 0
    y + 4 = 0
             y = −4     		OR y + 6 = 0
              y = −6

    Notice that it is possible to factor the left side of the equation first: 16 − y2 = (4 + y)(4 − y). However, doing so is potentially dangerous: You may decide to then divide both sides of the equation by (4 + y). You cannot do this, because it is possible that (4 + y) equals 0 (and, in fact, for one solution of the equation, it does).

  4. x = {−3, 3}
  5. x = {15, −2}


  6. s = 7

    The area of the square = s2. The perimeter of the square = 4s:

  7. t = 2
  8. 2

    Use FOIL to simplify this product:

  9. 19

    Factor both quadratic equations. Then use the greatest possible values of x and y to find the maximum value of the sum x + y:


    or
    or

    The maximum possible value of x + y = 9 + 10 = 19.

  10. x = {1, 9}

    or
  11. (D)

    First, factor the equation in the common information:

    x2 − 2x − 15 = 0 → (x − 5)(x + 3) = 0
    x = 5 or x = −3

    x2 − 2x − 15 = 0

    Quantity A Quantity B
    x = 5 or –3 1

    The value of x could be greater than or less than 1. The relationship cannot be determined.

  12. (C)

    First, factor the equation in the common information:

    x2 − 12x + 36 = 0 → (x − 6)(x − 6) = 0
    x = 6

    x2 – 12x + 36 = 0
    Quantity A Quantity B
    x = 6 6

    The two quantities are equal.

  13. (A)

    Expand the expressions in both columns:

    xy>0
    Quantity A Quantity B

    Now subtract x2 + y2 from both columns:

    xy>0
    Quantity A Quantity B

    Because xy is positive, Quantity A will be positive, regardless of the values of x and y. Similarly, Quantity B will always be negative, regardless of the values of x and y.

    Quantity A is greater.