Kindly help me to see whether my presentation to both questions are correct thanks
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Question
Answer
Your Question 4 solution does not match the Question 4 in your paper.
Question 4
Show that -x^{2} +2x – 4 is always negative for all real value of x.
In order for a quadratic function to be always negative, it must be
- b²-4ac < 0
- a < 0
for all real values of x.
a =-1, b = – 2, c = -4
a < 0. second condition is satisfied.
b²-4ac = (-2)^{2} – 4(-1)(-4) = 4 – 16 = -12 < 0 first condition is satisfied
Hence -x^{2} +2x – 4 is always negative for all real value of x.
Question 5:
Show that kx^{2} + (k-1)x + (1-2k) = 0 where k <> 0, has real roots for all values of k.
For real roots, b²-4ac ≥ 0
a = k, b = (k-1), c = (1-2k)
(k-1)^{2} – 4(k)(1-2k)
k^{2} – 2k + 1 – 4k +8k^{2 }
9k^{2} -6k + 1 = (3k-1)(3k-1)
This means (3k-1)^{2} ≥ 0 for all values of k and that the quadratic equation has real roots.
3k ≥ 1
k ≥ (1/3) i.e. k <> 0
So for all k ≥ (1/3) gives kx^{2} + (k-1)x + (1-2k) = 0 real roots and k <> 0.
Thanks very much.
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