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Score: \( 0 / 4 \) Penalty: 1 off Question Solve the equation for all values of \( x \) \[ 3 x(8 x-9)\left(x^{2}+64\right)=0 \]
Ask by Tucker Hart. in the United States
Feb 02,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solutions are \( x=0 \) and \( x=\frac{9}{8} \).
Solution
Solve the equation \( 3x(8x-9)(x^{2}+64)=0 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(3x\left(8x-9\right)\left(x^{2}+64\right)=0\)
- step1: Calculate:
\(x\left(8x-9\right)\left(x^{2}+64\right)=0\)
- step2: Separate into possible cases:
\(\begin{align}&x=0\\&8x-9=0\\&x^{2}+64=0\end{align}\)
- step3: Solve the equation:
\(\begin{align}&x=0\\&x=\frac{9}{8}\\&x \notin \mathbb{R}\end{align}\)
- step4: Find the union:
\(\begin{align}&x=0\\&x=\frac{9}{8}\end{align}\)
- step5: Rewrite:
\(x_{1}=0,x_{2}=\frac{9}{8}\)
The solutions to the equation \(3x(8x-9)(x^{2}+64)=0\) are \(x=0\) and \(x=\frac{9}{8}\).
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Simplify this solution Extra Insights
To solve the equation \( 3 x(8 x - 9)(x^2 + 64) = 0 \), we need to set each factor equal to zero. 1. First factor: \( 3x = 0 \) - This gives us \( x = 0 \). 2. Second factor: \( 8x - 9 = 0 \) - Solving this, we get: \[ 8x = 9 \\ x = \frac{9}{8} \] 3. Third factor: \( x^2 + 64 = 0 \) - Rearranging gives us: \[ x^2 = -64 \] - Since the square of a real number cannot be negative, this does not yield any real solutions. Instead, we find complex solutions: \[ x = \pm 8i \] Combining all the solutions, we have: \[ x = 0, \quad x = \frac{9}{8}, \quad x = 8i, \quad x = -8i \] So, the complete solution set is: \[ \{0, \frac{9}{8}, 8i, -8i\} \]
