Question

solve the equation. if necessary, round to the nearest hundredth.\\(x^3 - 6x^2 + 4x = -6\\)\\(\bigcirc\\) a) \\(-0.69, 1.75, 4.95\\)\\(\bigcirc\\) b) \\(-0.69, -1.75, 4.95\\)\\(\bigcirc\\) c) \\(-0.69, 4.95\\)\\(\bigcirc\\) d) no solution

Explanation:

Step1: Rearrange equation to standard form

$x^3 - 6x^2 + 4x + 6 = 0$

Step2: Use rational root theorem (no rational roots)

Test $\pm1,\pm2,\pm3,\pm6$; none satisfy the equation.

Step3: Apply numerical root-finding method

Use Newton-Raphson method or calculator to approximate roots:

• First root: $x \approx -0.69$
• Second root: $x \approx 1.75$
• Third root: $x \approx 4.95$

Step4: Verify roots in original equation

Substitute each root into $x^3 - 6x^2 + 4x$; results are $\approx -6$.

Answer:

A) -0.69, 1.75, 4.95