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find all a (a is any real number) for wich the equation a^3+ a^2|a+x| + |a^2x+1| = 1 has not less than four different solutions wich are integers

find all a (a is any real number) for wich the equation a^3+ a^2|a+x| + |a^2x+1| = 1 has not less than four different solutions wich are integers

Grade:12

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
7 years ago
Ans:
Hello Student,
Please find answer to your question below

a^3 + a^2|a+x| + |a^2x+1| = 1
|a+x| = a+x; x\geq -a
|a+x| = -a-x; x\leq -a
|a^2x+1| = a^2x+1;x\geq \frac{-1}{a^2}
|a^2x+1| = -a^2x-1;x\leq \frac{-1}{a^2}
For a > 0
-a\leq x\leq \frac{-1}{a^2}
a^3 + a^2|a+x| + |a^2x+1| = 1
a^3 + a^2(a+x) - (a^2x+1) = 1
2a^3 = 2
a^3 = 1
\Rightarrow a = 1
x\leq -a
a^3 + a^2|a+x| + |a^2x+1| = 1
a^3 - a^2(a+x) - (a^2x+1) = 1
-2a^{2}x = 2
-a^{2}x = 1
a^{2} = \frac{-1}{x}
a = \sqrt{\frac{-1}{x}}
x\geq \frac{-1}{a^2}
a^3 + a^2|a+x| + |a^2x+1| = 1
a^3 + a^2(a+x) + (a^2x+1) = 1
a^3 + a^2x = 0
a^2(a + x) = 0
a = 0
a = -x;x\leq 0

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