+223 Of the five quadratics listed below, four of them have two distinct roots. The fifth quadratic has a repeated root. Find the value of the repeated root.
\(\begin{align*} &-x^2 + 18x + 81 \\ &3x^2 - 3x - 168 \\ &x^2 - 4x - 4 \\ &25x^2 - 30x + 9 \\ &x^2 - 14x + 24 \end{align*}\)
+128460 We can only have a repeated root whenever the discriminant = 0
First one 18^2 - 4(-1)(81) > 0 so no repeated root
Second one (-3)^2 - 4(3) (-168) > 0 so no repeated root
Third one (-4) - (4) (1) (-4) > 0 so no repeated root
Fourth one (-30)^2 - 4(25) (9) = 900 - 100(9) = 0 repeated root
Note this factors as (5x - 3)^2
(5x - 3)^2 = 0 take both roots
5x - 3 = 0
5x = 3
x = 3/5 = repeated root
