Question

if x=-5+2root-4 , find the value of x⁴+9x³+35x²-x+4

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Answer 1

Answer:

x = -5 + 2√(-4)

x = -5 + 4i

Now,

x² = (-5 + 4i).(-5 + 4i) = 9 - 40i

x³ = (9 - 40i).(-5 + 4i) = 115 + 236i

x⁴ = (115 + 236i).(-5 + 4i) = -1519 - 720i

Therefore,

x⁴ + 9x³ + 35x² - x + 4

= -1519 - 720i + 9(115 + 236i) + 35(9 - 40i) - (-5 + 4i) + 4

= -1519 - 720i + 1035 + 2124i + 315 - 1400i + 5 - 4i + 4

= (-1519 + 1035 + 315 + 5 +4) + (-720 + 2124 - 1400 - 4)i

= (-160) + (0)i

= -160

Thus, the value of x⁴ + 9x³ + 35x² - x + 4 is -160.

Answer 2

Answer:

-160

Step-by-step explanation:

Given: x = 5 + 2√-4

⇒ x - 5 = 2√-4

On Squaring both sides, we get

⇒ (x + 5)² = (2√-4)²

⇒ x² + 25 + 10x = (2 * 2i)²

⇒ x² + 25 + 10x = 16i²

⇒ x² + 25 + 10x = -16[∵ i² = -1]

⇒ x² + 10x + 41 = 0

Now,

Given Equation is x⁴ + 9x³ + 35x² - x + 4.

= x⁴ + 10x³ - x³ + 41x² - 10x² + 4x² - 41x + 40x + 164 - 160

= x⁴ + 10x³ + 41x² - x³ - 10x² - 41x + 4x² + 40x + 164 - 160

= x²(x² + 10x + 41) - x(x² + 10x + 41) + 4(x² + 10x + 41) - 160

= x²(0) - x(0) + 4(0) - 160

= -160

Hope it helps!