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The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equation
p(p(x)) = 0 has
p(p(x)) = 0 has
- a)only purely imaginary roots
- b)all real roots
- c)two real and two purely imaginary roots
- d)neither real nor purely imaginary roots
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The quadratic equation p(x) = 0 with real coefficients has purely imag...
Hence real or purely imaginary number can not satisfy P(P(x)) = 0.
Most Upvoted Answer
The quadratic equation p(x) = 0 with real coefficients has purely imag...
Explanation:
To understand why the correct answer is option 'D', let's analyze the given information step by step.
1. The quadratic equation p(x) = 0 has purely imaginary roots.
A quadratic equation with real coefficients can be written in the form:
p(x) = ax^2 + bx + c = 0
where a, b, and c are real numbers.
If the quadratic equation has purely imaginary roots, it means that the discriminant of the equation is negative. The discriminant is given by:
D = b^2 - 4ac
If D < 0,="" the="" roots="" of="" the="" equation="" will="" be="" purely="" />
2. p(p(x)) = 0
Now, let's consider the equation p(p(x)) = 0. To find the roots of p(p(x)), we need to find the values of x that satisfy this equation.
Let's substitute p(x) with y:
p(y) = 0
Now, we have the equation p(y) = 0, which is a quadratic equation. We know that this equation has purely imaginary roots.
3. Analyzing p(y) = 0
Since p(y) = 0 has purely imaginary roots, the discriminant of p(y) must be negative:
D' = b'^2 - 4a'c' < />
where a', b', and c' are the coefficients of the quadratic equation p(y) = 0.
4. Analyzing p(p(x)) = 0
Let's substitute y back with p(x) in the discriminant equation:
D' = b'^2 - 4a'c' = (b(p(x)))^2 - 4(a(p(x))(c(p(x))))
Since p(y) = 0 has purely imaginary roots, we know that D' < 0.="" therefore,="" the="" discriminant="" of="" p(p(x))="0" is="" />
5. Conclusion
From our analysis, we can conclude that the equation p(p(x)) = 0 has no real roots or purely imaginary roots. Therefore, the correct answer is option 'D' - neither real nor purely imaginary roots.
This conclusion is based on the fact that if the quadratic equation p(x) = 0 has purely imaginary roots, then any composition of p(x) will also have neither real nor purely imaginary roots.
To understand why the correct answer is option 'D', let's analyze the given information step by step.
1. The quadratic equation p(x) = 0 has purely imaginary roots.
A quadratic equation with real coefficients can be written in the form:
p(x) = ax^2 + bx + c = 0
where a, b, and c are real numbers.
If the quadratic equation has purely imaginary roots, it means that the discriminant of the equation is negative. The discriminant is given by:
D = b^2 - 4ac
If D < 0,="" the="" roots="" of="" the="" equation="" will="" be="" purely="" />
2. p(p(x)) = 0
Now, let's consider the equation p(p(x)) = 0. To find the roots of p(p(x)), we need to find the values of x that satisfy this equation.
Let's substitute p(x) with y:
p(y) = 0
Now, we have the equation p(y) = 0, which is a quadratic equation. We know that this equation has purely imaginary roots.
3. Analyzing p(y) = 0
Since p(y) = 0 has purely imaginary roots, the discriminant of p(y) must be negative:
D' = b'^2 - 4a'c' < />
where a', b', and c' are the coefficients of the quadratic equation p(y) = 0.
4. Analyzing p(p(x)) = 0
Let's substitute y back with p(x) in the discriminant equation:
D' = b'^2 - 4a'c' = (b(p(x)))^2 - 4(a(p(x))(c(p(x))))
Since p(y) = 0 has purely imaginary roots, we know that D' < 0.="" therefore,="" the="" discriminant="" of="" p(p(x))="0" is="" />
5. Conclusion
From our analysis, we can conclude that the equation p(p(x)) = 0 has no real roots or purely imaginary roots. Therefore, the correct answer is option 'D' - neither real nor purely imaginary roots.
This conclusion is based on the fact that if the quadratic equation p(x) = 0 has purely imaginary roots, then any composition of p(x) will also have neither real nor purely imaginary roots.
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The quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then the equationp(p(x)) = 0 hasa)only purely imaginary rootsb)all real rootsc)two real and two purely imaginary rootsd)neither real nor purely imaginary rootsCorrect answer is option 'D'. Can you explain this answer?
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