Sum and product of the roots

Statement

Without solving it, find the sum and the product of the roots of the equation $4x^2 - 20x + 9 = 0$.

Solution

For an equation $ax^2 + bx + c = 0$ with roots $x_1, x_2$, Vieta’s formulas hold: $x_1 + x_2 = -\frac{b}{a}, \qquad x_1 \cdot x_2 = \frac{c}{a}.$ With $a=4$, $b=-20$, $c=9$: $x_1 + x_2 = -\frac{-20}{4} = 5, \qquad x_1 \cdot x_2 = \frac{9}{4} = 2.25.$ Sanity check: the discriminant is $400-144 = 256 > 0$, so the roots are real (in fact $\tfrac12$ and $\tfrac92$, with sum $5$ and product $\tfrac94$). $\boxed{x_1 + x_2 = 5, \qquad x_1 x_2 = \tfrac94}$