Question

3.
constant
The equation x² +px+q = 0, where p and q are constants, has roots -3 and 5.
(i) Find the values of p and q.
(ii) Using these values of p and q, find the value of the constant r for which the equation
x² +px+q+r=0 has equal roots.

Answer 1

(i) Use the fundamental theorem of algebra (or Vieta's formulas).

$x^2 + px + q = (x + 3) (x - 5) = x^2 - 2x - 15 \\\\ \implies \boxed{p = -2 \text{ and } q = -15}$

(ii) By completing the square,

$x^2 + px + q + r = \left(x + \dfrac p2\right)^2 + q + r - \dfrac{p^2}4$

Then using the values of $p,q$, we have

$(x - 1)^2 - 15 + r - 1 = 0 \implies (x - 1)^2 = 16 - r$

Observe that if $\boxed{r=16}$, then the quadratic has two roots at $x=1$, since this would give

$(x - 1)^2 = 0$