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Advanced Quadratic Problem Set

1. Evaluate 2a5 - 5a4 + 2a3 - 8a2 / a2 + 1, given that a is a root of x2 - 3x + 1 = 0. 2. Find the value of b such that the equations 1988x2 + bx + 8891 = 0 and 8891x2 + bx + 1988 = 0 have a common root. 3. Find the value of -b/(a + b - a), given that the equations a - 1x2 - a2 + 2x + a2 + 2a = 0 and b - 1x2 - b2 + 2x + b + 2b = 0 have one common root, where a

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578 views2 pages

Advanced Quadratic Problem Set

1. Evaluate 2a5 - 5a4 + 2a3 - 8a2 / a2 + 1, given that a is a root of x2 - 3x + 1 = 0. 2. Find the value of b such that the equations 1988x2 + bx + 8891 = 0 and 8891x2 + bx + 1988 = 0 have a common root. 3. Find the value of -b/(a + b - a), given that the equations a - 1x2 - a2 + 2x + a2 + 2a = 0 and b - 1x2 - b2 + 2x + b + 2b = 0 have one common root, where a

Uploaded by

Erwin Ello
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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LATIHAN 7

1 Given that a is a root of equation x 2 − 3x + 1 = 0, evaluate

2a5 − 5a4 + 2a3 − 8a2


a2 + 1

2 For what value of b do the equations ∶ 1988x 2 + bx + 8891 = 0


and 8891x 2 + bx + 1988 = 0 have a common root?

3 a, b are two different positive integers, and the two quadratic equations
a − 1 x 2 − a2 + 2 x + a2 + 2a = 0 and b − 1 x 2 − b2 + 2 x + b + 2b = 0

ab + ba
have one common root. Find the value of −b .
a + b −a

4 Find the value of k, such that the equations x 2 − kx − 7 = 0 and


x 2 − 6x − k + 1 = 0 have a common root. Find the common root and
different roots.

5 Solve the quadratic inequality ax 2 − a + 1 x + 1 < 0, where a is a parameter.

6 Given that the inequality kx 2 − kx − 1 < 0 holds for any real x.


Find the range of k.

7 Given that the solution set of the quadratic inequality ax 2 + bx + c > 0


is 1 < x < 2. Find the solution set of the inequality cx 2 + bx + a < 0
8 Given that the quadratic function f x = x 2 − 2ax + 6 ≥ a for − 2 ≤ x ≤ 2,
find the range of the constant a.

1
9 Given that the inequality 2a − a2 ≤ x 2 − 3x + 2 ≤ 3 − a2
8
holds for any real x in the interval 0, 2 . Find the range of a parameter a.

10 Given that the equation x 2 + 2a − 1 x + a2 = 0 has two real positive roots,


where a is an integer. If x1 and x2 are the roots, find the value of x1 − x2 .

11 x1 and x2 are roots of the equation x 2 + x − 3 = 0. Find the value of x13 − 4x22 + 19.

12 If a, b are real number and a2 + 3a + 1 = 0, b2 + 3b + 1 = 0,

a b
find the value of + .
b a

13 Given that a, b are integers with a > b, and the two roots α, β of the equation
3x 2 + 3 a + b x + 4ab = 0 satisfy the relation
α α + 1 + β β + 1 = α + 1 β + 1 , find all pairs a, b of two integers.

14 If p, q are two real numbers satisfying the relations 2p2 − 3p − 1 = 0 and

pq + p + 1
q2 + 3q − 2 = 0 and pq ≠ 1. Find the value of .
q

15 Given that the real numbers s, t satisfy 19s 2 + 99s + 1 = 0,

st + 4s + 1
t 2 + 99t + 19 = 0, and st ≠ 1. Find the value of .
t

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