F.4 Quadratic Equations

1)
The graph of y = x² + mx+ 16 has one x-intercept,
then the equation x² + mx+ 16 = 0 has only one real root.

Discriminant Δ = 0
m² - 4(1)(16) = 0
m² = 64
m = 8 or m= -8


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2)
(a)
x² + (5 - 2k)x + (k – 1) = 0
Sum of the roots : α + β = -(5 - 2k) = 2k - 5
Product of the roots : αβ = k - 1

2(α + β) =αβ
2(2k - 5) = k - 1
4k - 10 = k - 1
3k = 9
k = 3

(b)
x² + (5 - 2k)x + (k – 1) = 0
k = 3, then x² + [5 - 2(3)]x + [(3)– 1] = 0
x² - x + 2 = 0

Discriminant Δ
= (-1)² - 4(1)(2)
= -7 < 0

The discriminant of the equation < 0
The roots α and β are NOT realnumbers.