Question:
If
\[
x=-5+2\sqrt{-4},
\]
then the value of
\[
x^4+9x^3+35x^2-x+4
\]
is:
If
\[
x=-5+2\sqrt{-4},
\]
then the value of
\[
x^4+9x^3+35x^2-x+4
\]
is:
Show Hint
When a complex number satisfies a quadratic equation, reduce higher powers using the quadratic relation instead of expanding directly.
Updated On: Jun 24, 2026
- \(80\)
- \(160\)
- \(-160\)
- \(-80\)
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Simplify the given value of \(x\).
Since \[ \sqrt{-4}=2i, \] we get \[ x=-5+4i \] Observe that \[ x+5=4i \] Squaring both sides, \[ (x+5)^2=(4i)^2 \] \[ x^2+10x+25=-16 \] \[ x^2+10x+41=0 \] Thus, \[ x^2=-10x-41 \]
Step 2: Find \(x^3\).
Multiply by \(x\): \[ x^3=x(-10x-41) \] \[ x^3=-10x^2-41x \] Substitute \[ x^2=-10x-41 \] \[ x^3=-10(-10x-41)-41x \] \[ x^3=100x+410-41x \] \[ x^3=59x+410 \]
Step 3: Find \(x^4\).
Multiply by \(x\): \[ x^4=x(59x+410) \] \[ x^4=59x^2+410x \] Again substitute \[ x^2=-10x-41 \] \[ x^4=59(-10x-41)+410x \] \[ x^4=-590x-2419+410x \] \[ x^4=-180x-2419 \]
Step 4: Evaluate the expression.
Now, \[ x^4+9x^3+35x^2-x+4 \] Substitute all values: \[ =(-180x-2419)+9(59x+410)+35(-10x-41)-x+4 \] \[ =-180x-2419+531x+3690-350x-1435-x+4 \] Combine \(x\)-terms: \[ (-180+531-350-1)x=0 \] Combine constants: \[ -2419+3690-1435+4 \] \[ =1271-1431 \] \[ =-160 \]
Step 5: Final conclusion.
Hence, \[ \boxed{-160} \]
Since \[ \sqrt{-4}=2i, \] we get \[ x=-5+4i \] Observe that \[ x+5=4i \] Squaring both sides, \[ (x+5)^2=(4i)^2 \] \[ x^2+10x+25=-16 \] \[ x^2+10x+41=0 \] Thus, \[ x^2=-10x-41 \]
Step 2: Find \(x^3\).
Multiply by \(x\): \[ x^3=x(-10x-41) \] \[ x^3=-10x^2-41x \] Substitute \[ x^2=-10x-41 \] \[ x^3=-10(-10x-41)-41x \] \[ x^3=100x+410-41x \] \[ x^3=59x+410 \]
Step 3: Find \(x^4\).
Multiply by \(x\): \[ x^4=x(59x+410) \] \[ x^4=59x^2+410x \] Again substitute \[ x^2=-10x-41 \] \[ x^4=59(-10x-41)+410x \] \[ x^4=-590x-2419+410x \] \[ x^4=-180x-2419 \]
Step 4: Evaluate the expression.
Now, \[ x^4+9x^3+35x^2-x+4 \] Substitute all values: \[ =(-180x-2419)+9(59x+410)+35(-10x-41)-x+4 \] \[ =-180x-2419+531x+3690-350x-1435-x+4 \] Combine \(x\)-terms: \[ (-180+531-350-1)x=0 \] Combine constants: \[ -2419+3690-1435+4 \] \[ =1271-1431 \] \[ =-160 \]
Step 5: Final conclusion.
Hence, \[ \boxed{-160} \]
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