Solve the equation \( 6 x^{\frac{1}{3}}-2 x^{-\frac{1}{3}}-1=0 \). Give your answers in exact form.
Solución de inteligencia artificial de Upstudy
Responder
Solución
Respondido por UpStudy AI y revisado por un tutor profesional
Explicar 
Simplifique esta solución Mind Expander
To solve the equation \( 6 x^{\frac{1}{3}} - 2 x^{-\frac{1}{3}} - 1 = 0 \), we start by rewriting the equation for ease of manipulation. Let's set \( y = x^{\frac{1}{3}} \). Therefore, \( x^{-\frac{1}{3}} = \frac{1}{y} \). Substituting these into the equation gives: \[ 6y - 2 \cdot \frac{1}{y} - 1 = 0 \] To eliminate the fraction, we can multiply through by \( y \) (assuming \( y \neq 0 \)): \[ 6y^2 - 2 - y = 0 \] Rearranging this yields: \[ 6y^2 - y - 2 = 0 \] Now, we can apply the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 6 \), \( b = -1 \), and \( c = -2 \): \[ b^2 - 4ac = (-1)^2 - 4 \cdot 6 \cdot (-2) = 1 + 48 = 49 \] Now substituting this back into the quadratic formula: \[ y = \frac{-(-1) \pm \sqrt{49}}{2 \cdot 6} = \frac{1 \pm 7}{12} \] This results in two potential solutions for \( y \): 1. \( y = \frac{8}{12} = \frac{2}{3} \) 2. \( y = \frac{-6}{12} = -\frac{1}{2} \) Since \( y = x^{\frac{1}{3}} \), we need to find \( x \) corresponding to each valid \( y \). For \( y = \frac{2}{3} \): \[ x^{\frac{1}{3}} = \frac{2}{3} \] Cubing both sides, we obtain: \[ x = \left(\frac{2}{3}\right)^3 = \frac{8}{27} \] For \( y = -\frac{1}{2} \): \[ x^{\frac{1}{3}} = -\frac{1}{2} \] Cubing both sides leads to: \[ x = \left(-\frac{1}{2}\right)^3 = -\frac{1}{8} \] Thus, the exact solutions for the equation \( 6 x^{\frac{1}{3}} - 2 x^{-\frac{1}{3}} - 1 = 0 \) are: \[ x = \frac{8}{27} \quad \text{and} \quad x = -\frac{1}{8} \]
