Multiple choice

Y = 2(x-a)(x-b) In the quadratic equation above, a and b are positive constants, with a > b. If the graph of the equation in the xy- plane is a parabola with vertex (6, -18), what is the value of a?

  1. 3

  2. 6

  3. 9

  4. 12
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Vertex form is y = 2(x - 6)^2 - 18. Expanding 2(x - a)(x - b) = 2(x^2 - (a+b)x + ab). Comparing coefficients, a+b = 12 and ab = 6 + 9 = 15. Solving x^2 - 12x + 27 = 0 gives roots 9 and 3. Since a > b, a = 9.

AI explanation

The axis of symmetry for the parabola is halfway between the x-intercepts a and b, giving (a + b) / 2 = 6, which simplifies to a + b = 12. The vertex represents the maximum value since the leading coefficient is positive, so substituting x = 6 and Y = -18 gives -18 = 2(6 - a)(6 - b). Dividing by 2 yields -9 = (6 - a)(6 - b), which expands to -9 = 36 - 6(a + b) + ab; substituting a + b = 12 results in -9 = 36 - 72 + ab, meaning ab = 27. Solving the system a + b = 12 and ab = 27 gives the quadratic t^2 - 12t + 27 = 0, whose roots are 3 and 9; since a > b, a must be 9.

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