13. For any quadratic equation to have imaginary roots, b² – 4ac must be less than 0
b² – 4ac < 0
a = 2 ; b = n ; c = 5
n² – 4(2)(5) < 0
n² – 40 < 0
n² < 40
=> −40<n<40
= −210<n<210
14. Let y=x²+1x+1
Make x the subject
yx²+y=x+1
yx²−x+y−1=0
From the general quadratic formula
ax²+bx+c=0
We can say that
a = y ; b = -1; c = y-1
from the quadratic formula
x=2a−b±b²−4ac
x=2y−(−1)±(−1)²−4(y)(y−1)
x=2y1±1−4(y²−y
x=2y1±1−4y²+4y
For the above equation to be defined, then 1−4y²+4y≥0
-4y² + 4y + 1 ≥ 0
4y² - 4y - 1 ≤ 0
Divide through by 4
y²−y−41≤0
y²−y+41−41−41≤0
y²−y+41−21≤0
Multiply through by 4
(4y² – 4y + 1) – 2 ≤ 0
(2y – 1)² – 2 ≤ 0
(2y – 1)² ≤ 2
=> −2≤2y−1≤2
−2≤2y−1
2y−1≥−2
y≥21−2
=>2y−1≤2
y≤22+1
The range of values is
[21−2,0)⋃(0,22+1]
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