Question

solve the equation. if necessary, round to the nearest hundredth.\\(x^4 + 2x^3 = 7\\)\\(\bigcirc\\) a) \\(-2.47, 1.29, 2.47\\)\\(\bigcirc\\) b) \\(-1.29, 2.47, 1.29\\)\\(\bigcirc\\) c) \\(-2.47, 1.29\\)\\(\bigcirc\\) d) no solution

Explanation:

Step1: Rearrange the equation

Rewrite as $x^4 + 2x^3 - 7 = 0$

Step2: Analyze polynomial degree

A quartic has max 4 real roots. Test sign changes:

• For $x\to+\infty$: $f(x)\to+\infty$; $f(1)=1+2-7=-4$; $f(2)=16+16-7=25$. Sign change between 1&2: root ~1.29.
• For $x\to-\infty$: $f(x)\to+\infty$; $f(-2)=16-16-7=-7$; $f(-3)=81-54-7=20$. Sign change between -3&-2: root ~-2.47.
• $f(-1)=1-2-7=-8$; no sign change between -2&-1, so no real roots here.

Step3: Match to options

Only 2 real roots: -2.47, 1.29

Answer:

C) $-2.47, 1.29$