us The quadratic functions f and g are defined by f(x) = 4x² +p, XER - = g(x) = x² +qx + p, x-ER where p, q and r are non-zero constants such that p =-2r and q=-3r. Given that (x +r) is a factor of f and g. Determine the value of p, q and r. Hence, factorise f and g completely.

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Step-by-step explanation:

Substituting for p in the equation for f(x), we can use synthetic division to find the value of r that makes the remainder zero from division by (x +r).

To make (x+r) be a factor, the remainder (4r² -2r) must be zero. That factors as ...

  2r(2r -1) = 0   ⇒   r = 1/2; r = 0 is disallowed by the problem statement

Then p = -2r = -1, and q = -3r = -3/2.

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The functions with the values filled in are ...

  f(x) = 4x² -1 = (2x -1)(2x +1)

  g(x) = x² -3/2x -1 = (x -2)(x +1/2)

For this case we have the following quadratic equation:

Applying the quadratic formula we have:

Where:

Substituting we have:

We have two roots:

Option B