F.4 Mathematics Quadratic

2009-10-03 9:35 pm

1. It is given that 3A and 3B are the roots of the quadratic equation
x^2 + px + q = 0. Form a quadratic equation in x whose roots are A + 2B and B + 2A, in terms of p and q.

2. Prove that the quadratic equation ( x-k)[x-(k+1)] =1 has two distinct real roots for any real values of k.

3. It is given that the quadratic equation px^2 + qx + r =0 has two
distinct real roots.
(a) Determine whether the quadratic equation px^2 + 2qx + r = 0 has
distinct real roots. Explain your answer.
(b) By investigating the result of (a), suggest two quadratic equations
in x which have two distinct real roots, in terms of p,q and r.

回答 (1)

myisland8132

2009-10-03 9:40 pm

✔ 最佳答案

1 3(A+B)=-p,9AB=q
A+2B+B+2A=3(A+B)=-p
(A+2B)(B+2A)=2(A^2+B^2)+5AB=2[(A+B)^2]+AB
=2p^2+q
The required quadratic equation is
x^2+px+(2p^2+q)=0
2 ( x-k)[x-(k+1)] =1
x^2-(2k+1)x+k(k+1)-1=0
D=(2k+1)^2-4(k^2+k-1)
=4k^2+4k+1-4k^2-4k+4
=5 > 0
So, the quadratic equation ( x-k)[x-(k+1)] =1 has two distinct real roots for any real values of k.
3 (a) We have q^2-4pr>0
Consider D=4q^2-4pq=3q^2+(q^2-4pr)>0
So, the quadratic equation px^2 + 2qx + r = 0 has distinct real roots.
(b) px^2 + 3qx + r = 0, px^2 + 4qx + r = 0

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