Question
Solve for \( x \) \( 6 x^{3}-5 x^{2}-3 x+2=0 \) \( x=\frac{[?]}{[]}, \frac{[]}{\square]},[\square \) Enter the smallest answer first.
Ask by Olson Powers. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solutions are \(x = -\frac{2}{3}\), \(x = \frac{1}{2}\), and \(x = 1\).
Solution
Solve the equation \( 6x^{3}-5x^{2}-3x+2=0 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(6x^{3}-5x^{2}-3x+2=0\)
- step1: Factor the expression:
\(\left(x-1\right)\left(2x-1\right)\left(3x+2\right)=0\)
- step2: Separate into possible cases:
\(\begin{align}&x-1=0\\&2x-1=0\\&3x+2=0\end{align}\)
- step3: Solve the equation:
\(\begin{align}&x=1\\&x=\frac{1}{2}\\&x=-\frac{2}{3}\end{align}\)
- step4: Rewrite:
\(x_{1}=-\frac{2}{3},x_{2}=\frac{1}{2},x_{3}=1\)
The solutions to the equation \(6x^{3}-5x^{2}-3x+2=0\) are:
1. \(x = -\frac{2}{3}\)
2. \(x = \frac{1}{2}\)
3. \(x = 1\)
Therefore, the solutions are \(x = -\frac{2}{3}, x = \frac{1}{2}, x = 1\).
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Simplify this solution Mind Expander
To solve the cubic equation \( 6x^3 - 5x^2 - 3x + 2 = 0 \), we can use methods such as synthetic division or the Rational Root Theorem to find potential rational roots. Testing possible rational roots, we discover that \( x = \frac{1}{3} \) is a solution. After factoring out \( (x - \frac{1}{3}) \), we can rewrite the equation and find the other roots. The complete solutions are \( x = \frac{2}{3}, -1, \) and \( \frac{1}{2} \). Arranging the solutions from smallest to largest, we have: \( x = -1, \frac{1}{2}, \frac{2}{3}\) Thus, the final answer is: \( x = [-1, \frac{1}{2}, \frac{2}{3}] \)
