Question

solve the equation. if necessary, round to the nearest hundredth.\\(x^4 - 15x^2 + x + 65 = 0\\)\\(\bigcirc\\) a) \\(-2.76, 0.03, 2.72\\)\\(\bigcirc\\) b) \\(-2.76, 2.72\\)\\(\bigcirc\\) c) \\(0.03\\)\\(\bigcirc\\) d) no solution

Explanation:

Step1: Test candidate x=-2.76

$(-2.76)^4 - 15(-2.76)^2 + (-2.76) + 65$
$=59.09 - 15(7.62) -2.76 +65$
$=59.09 - 114.3 -2.76 +65 \approx 6.93$
(Close to 0, considering rounding)

Step2: Test candidate x=2.72

$(2.72)^4 - 15(2.72)^2 + 2.72 + 65$
$=54.70 - 15(7.40) +2.72 +65$
$=54.70 - 111 +2.72 +65 \approx 11.42$
(Close to 0, considering rounding)

Step3: Test candidate x=0.03

$(0.03)^4 - 15(0.03)^2 + 0.03 + 65$
$\approx 0 - 15(0.0009) +0.03 +65$
$= -0.0135 +0.03 +65 = 65.0165
eq 0$

Step4: Check for real solutions

Analyze $f(x)=x^4-15x^2+x+65$. For large $|x|$, $x^4$ dominates, so $f(x)\to+\infty$. Find critical points by $f'(x)=4x^3-30x+1$. Testing values shows the function's minimum values are positive, meaning no real roots.

Answer:

D) no solution