Question
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3. Solve for \( x \) for the following: \( \quad(7 \) marks) \( 2 x^{3}+11 x^{2}-60 x+27=0 \)

Ask by Pritchard Wyatt. in Canada
Jan 24,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

The solutions are \(x=-9\), \(x=\frac{1}{2}\), and \(x=3\).

Solution

Solve the equation \( 2x^3+11x^2-60x+27=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(2x^{3}+11x^{2}-60x+27=0\) - step1: Factor the expression: \(\left(x-3\right)\left(x+9\right)\left(2x-1\right)=0\) - step2: Separate into possible cases: \(\begin{align}&x-3=0\\&x+9=0\\&2x-1=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=3\\&x=-9\\&x=\frac{1}{2}\end{align}\) - step4: Rewrite: \(x_{1}=-9,x_{2}=\frac{1}{2},x_{3}=3\) The solutions to the equation \(2x^{3}+11x^{2}-60x+27=0\) are \(x=-9\), \(x=\frac{1}{2}\), and \(x=3\).

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The Deep Dive

To solve the equation \( 2x^3 + 11x^2 - 60x + 27 = 0 \), we can start by using synthetic division or the Rational Root Theorem to find potential rational roots. Testing small integer values, we find \( x = 3 \) works because: \[ 2(3^3) + 11(3^2) - 60(3) + 27 = 54 + 99 - 180 + 27 = 0 \] Now that we have a root, we can perform synthetic division with \( x - 3 \) to factor the polynomial further: ``` 3 | 2 11 -60 27 | 6 51 -27 --------------------- 2 17 -9 0 ``` This gives us \( 2x^2 + 17x - 9 = 0 \). We can now use the quadratic formula to find the remaining roots: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-17 \pm \sqrt{17^2 - 4(2)(-9)}}{2(2)} = \frac{-17 \pm \sqrt{289 + 72}}{4} = \frac{-17 \pm \sqrt{361}}{4} \] Calculating \( \sqrt{361} = 19 \): \[ x = \frac{-17 \pm 19}{4} \] This gives us two solutions: 1. \( x = \frac{2}{4} = \frac{1}{2} \) 2. \( x = \frac{-36}{4} = -9 \) Thus, the solutions for \( x \) are: \[ x = 3, \quad x = \frac{1}{2}, \quad x = -9 \]

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