Pregunta
d) \( 8 x^{3}+4 x^{2}=2 x+1 \)
Ask by Alexander Mccoy. in Tunisia
Feb 17,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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Les solutions de l'équation sont \(x=-\frac{1}{2}\) et \(x=\frac{1}{2}\).
Solución
Solve the equation \( 8x^3+4x^2=2x+1 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(8x^{3}+4x^{2}=2x+1\)
- step1: Move the expression to the left side:
\(8x^{3}+4x^{2}-\left(2x+1\right)=0\)
- step2: Remove the parentheses:
\(8x^{3}+4x^{2}-2x-1=0\)
- step3: Factor the expression:
\(\left(2x-1\right)\left(2x+1\right)^{2}=0\)
- step4: Separate into possible cases:
\(\begin{align}&2x-1=0\\&\left(2x+1\right)^{2}=0\end{align}\)
- step5: Solve the equation:
\(\begin{align}&x=\frac{1}{2}\\&x=-\frac{1}{2}\end{align}\)
- step6: Rewrite:
\(x_{1}=-\frac{1}{2},x_{2}=\frac{1}{2}\)
La solution de l'équation \(8x^3+4x^2=2x+1\) est \(x=-\frac{1}{2}\) ou \(x=\frac{1}{2}\).
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Simplifique esta solución Beyond the Answer
To solve the equation \( 8 x^{3}+4 x^{2}-2 x-1 = 0 \), we can start by factoring or using numerical methods. Unfortunately, it's not straightforward to factor this polynomial directly, so we can apply the Rational Root Theorem to check for possible rational roots like \( x = 1, -1, \frac{1}{2}, \) etc. Plugging these values into the equation can lead us to potential solutions. Once we find a root, we can use synthetic division to simplify the polynomial further and continue solving the resulting quadratic equation. So don’t be shy – jump in and start testing values! You might just stumble upon the root! Keep your pen ready for synthetic division once you find a candidate!
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