Question

If you divide one of the roots of the equation 4x2+bx−27=0 by the other root, the quotient will be −3. Find b.

Answer 1

Answer:

  ±12   (two answers)

Explanation:

Suppose one root is a. Then the other root will be -3a. The product of the two roots is the ratio of the constant coefficient to the leading coefficient:

  (a)(-3a) = -27/4

  a² = -27/(4·(-3)) = 9/4

  a = ±√(9/4) = ±3/2

Then the other root is

  -3a = -3(±3/2) = ±9/2 . . . . . . the roots will have opposite signs

We know the opposite of the sum of these roots will be the ratio of the linear term coefficient to the leading coefficient: b/4, so ...

  -(a + (-3a)) = b/4

  2a = b/4

  b = 8a = 8·(±3/2)

  b = ±12

_____

Check

For b = 12, the equation factors as ...

  4x² +12x -27 = (2x -3)(2x +9) = 0

  It has roots -9/2 and +3/2, the ratio of which is -3.

For b = -12, the equation factors as ...

  4x² -12x -27 = (2x +3)(2x -9) = 0

  It has roots 9/2 and -3/2, the ratio of which is -3.