(a) Derive the smallest equation whose coefficients are integers and which has roots of \(\frac{1}{2}\) and -7. (b) Three years ago, a father was four times...

Answer Details

(a) If the roots are \(\tfrac{1}{2}\) and \(-7\), the equation is

\[ \left(x - \tfrac{1}{2}\right)(x + 7) = 0. \]

Multiply the first factor by 2 to clear the fraction (using \((2x-1)\)):

\[ (2x - 1)(x + 7) = 0 \implies 2x^2 + 14x - x - 7 = 0. \] \[ 2x^2 + 13x - 7 = 0. \]

This is the smallest equation with integer coefficients.

(b) Let the daughter's present age be \(d\) and the father's present age be \(f\).

"Three years ago the father was four times as old as the daughter is now":

\[ f - 3 = 4d \implies f = 4d + 3. \]

"The product of their present ages is 430":

\[ f \cdot d = 430 \implies (4d + 3)d = 430 \implies 4d^2 + 3d - 430 = 0. \]

Using the quadratic formula, \(\;d = \dfrac{-3 \pm \sqrt{3^2 + 4(4)(430)}}{2(4)} = \dfrac{-3 \pm \sqrt{6889}}{8} = \dfrac{-3 \pm 83}{8}.\)

Taking the positive root, \(d = \dfrac{80}{8} = 10\), so \(f = 4(10) + 3 = 43\).

The daughter is 10 years old and the father is 43 years old (product \(= 430\)).