Question:
Which of the following quadratic equations whose real roots \( x_1, x_2 \) satisfy the conditions \( x_1^2 + x_2^2 = 5 \), \( 3(x_1^5 + x_2^5) = 11(x_1^3 + x_2^3) \)?
Which of the following quadratic equations whose real roots \( x_1, x_2 \) satisfy the conditions \( x_1^2 + x_2^2 = 5 \), \( 3(x_1^5 + x_2^5) = 11(x_1^3 + x_2^3) \)?
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Roots and Identities}
Use symmetric sum identities for powers like \( x_1^2 + x_2^2, x_1^3 + x_2^3 \)
Derive expressions in terms of \( x_1 + x_2 \) and \( x_1 x_2 \)
Plug into given conditions to verify
Use symmetric sum identities for powers like \( x_1^2 + x_2^2, x_1^3 + x_2^3 \)
Derive expressions in terms of \( x_1 + x_2 \) and \( x_1 x_2 \)
Plug into given conditions to verify
Updated On: May 19, 2025
- \( x^2 + 3x + 2 = 0 \)
- \( x^2 + 3x + 11 = 0 \)
- \( x^2 + 5x + 2 = 0 \)
- \( x^2 + 5x + 11 = 0 \)
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The Correct Option is A
Solution and Explanation
Assume the roots of the quadratic are \( x_1 \) and \( x_2 \), then:
\[
x_1 + x_2 = -b, \quad x_1 x_2 = c.
\]
From the quadratic \( x^2 + 3x + 2 = 0 \), we get:
\[
x_1 + x_2 = -3, \quad x_1 x_2 = 2.
\]
Using identities:
\[
x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2 = 9 - 4 = 5.
\]
Also,
\[
x_1^3 + x_2^3 = (x_1 + x_2)^3 - 3x_1 x_2(x_1 + x_2) = (-3)^3 - 3(2)(-3) = -27 + 18 = -9,
\]
\[
x_1^5 + x_2^5 = (x_1 + x_2)(x_1^4 + x_2^4) - x_1 x_2(x_1^3 + x_2^3),
\]
which gives:
\[
3(-27) = 11(-9) \Rightarrow -81 = -81.
\]
Hence the quadratic \( x^2 + 3x + 2 = 0 \) satisfies all conditions.
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