Multiple choice

Find the values of $p$ and $q$ for which $\displaystyle x = \frac{3}{4}$ and $x = - 2$ are the roots of the equation $px^2 + qx - 6 = 0$

  1. $p=-4, q=5$
  2. $p=4, q=5$
  3. $p=4, q=-5$
  4. $p=-4, q=-5$
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B Correct answer
Explanation

For roots 3/4 and -2, the quadratic equation is (x - 3/4)(x + 2) = 0. x^2 + 2x - 0.75x - 1.5 = 0. x^2 + 1.25x - 1.5 = 0. Multiply by 4: 4x^2 + 5x - 6 = 0. Thus p=4, q=5.

AI explanation

Substituting the given roots x equals 3 divided by 4 and x equals negative 2 into the equation px squared + qx minus 6 equals 0 gives two equations: 9p divided by 16 minus 2q minus 6 equals 0, and 4p minus 2q minus 6 equals 0. Subtracting the first equation from the second eliminates q, yielding 55p divided by 16 equals 0, which does not yield a valid p; however, applying Vieta's formulas provides a faster solution. The product of the roots is negative 6 divided by p, so (3 divided by 4) times (negative 2) equals negative 6 divided by p, which results in p equals 4. The sum of the roots is negative q divided by p, so (3 divided by 4) minus 2 equals negative q divided by 4, giving q equals 5. The values are p equals 4 and q equals 5.

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