Pregunta
upstudy study bank question image url

5. \( f(x)=-2 x^{2}-x-2 \)

Ask by Chambers Mcdonald. in the United States
Mar 14,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

The function \( f(x) = -2x^2 - x - 2 \) is a downward-opening parabola with its vertex at \( \left(-\frac{1}{4}, -\frac{19}{8}\right) \) or \( \left(-0.25, -2.375\right) \). It has a y-intercept at \( (0, -2) \) and does not intersect the x-axis.

Solución

Evaluate the expression by following steps: - step0: Evaluate: \(-2x^{2}-x-2\) - step1: Substitute: \(-2\left(-\frac{1}{4}\right)^{2}-\frac{1}{4}-2\) - step2: Multiply the terms: \(-\frac{1}{8}-\frac{1}{4}-2\) - step3: Reduce fractions to a common denominator: \(-\frac{1}{8}-\frac{2}{4\times 2}-\frac{2\times 4\times 2}{4\times 2}\) - step4: Multiply the numbers: \(-\frac{1}{8}-\frac{2}{8}-\frac{2\times 4\times 2}{4\times 2}\) - step5: Multiply the numbers: \(-\frac{1}{8}-\frac{2}{8}-\frac{2\times 4\times 2}{8}\) - step6: Transform the expression: \(\frac{-1-2-2\times 4\times 2}{8}\) - step7: Multiply the terms: \(\frac{-1-2-16}{8}\) - step8: Subtract the numbers: \(\frac{-19}{8}\) - step9: Rewrite the fraction: \(-\frac{19}{8}\) Solve the equation \( -2x^{2}-x-2=0 \). Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(-2x^{2}-x-2=0\) - step1: Multiply both sides: \(2x^{2}+x+2=0\) - step2: Solve using the quadratic formula: \(x=\frac{-1\pm \sqrt{1^{2}-4\times 2\times 2}}{2\times 2}\) - step3: Simplify the expression: \(x=\frac{-1\pm \sqrt{1^{2}-4\times 2\times 2}}{4}\) - step4: Simplify the expression: \(x=\frac{-1\pm \sqrt{-15}}{4}\) - step5: Simplify the expression: \(x=\frac{-1\pm \sqrt{15}\times i}{4}\) - step6: Separate into possible cases: \(\begin{align}&x=\frac{-1+\sqrt{15}\times i}{4}\\&x=\frac{-1-\sqrt{15}\times i}{4}\end{align}\) - step7: Simplify the expression: \(\begin{align}&x=-\frac{1}{4}+\frac{\sqrt{15}}{4}i\\&x=\frac{-1-\sqrt{15}\times i}{4}\end{align}\) - step8: Simplify the expression: \(\begin{align}&x=-\frac{1}{4}+\frac{\sqrt{15}}{4}i\\&x=-\frac{1}{4}-\frac{\sqrt{15}}{4}i\end{align}\) - step9: Rewrite: \(x_{1}=-\frac{1}{4}-\frac{\sqrt{15}}{4}i,x_{2}=-\frac{1}{4}+\frac{\sqrt{15}}{4}i\) - step10: Remove the complex number(s): \(\textrm{No real solution}\) Solve the equation \( -2x^{2}-x-2=0 \). Solve the equation(The complex numbers system) by following steps: - step0: Solve using the quadratic formula in the complex numbers system: \(-2x^{2}-x-2=0\) - step1: Multiply both sides: \(2x^{2}+x+2=0\) - step2: Solve using the quadratic formula: \(x=\frac{-1\pm \sqrt{1^{2}-4\times 2\times 2}}{2\times 2}\) - step3: Simplify the expression: \(x=\frac{-1\pm \sqrt{1^{2}-4\times 2\times 2}}{4}\) - step4: Simplify the expression: \(x=\frac{-1\pm \sqrt{-15}}{4}\) - step5: Simplify the expression: \(x=\frac{-1\pm \sqrt{15}\times i}{4}\) - step6: Separate into possible cases: \(\begin{align}&x=\frac{-1+\sqrt{15}\times i}{4}\\&x=\frac{-1-\sqrt{15}\times i}{4}\end{align}\) - step7: Simplify the expression: \(\begin{align}&x=-\frac{1}{4}+\frac{\sqrt{15}}{4}i\\&x=\frac{-1-\sqrt{15}\times i}{4}\end{align}\) - step8: Simplify the expression: \(\begin{align}&x=-\frac{1}{4}+\frac{\sqrt{15}}{4}i\\&x=-\frac{1}{4}-\frac{\sqrt{15}}{4}i\end{align}\) - step9: Rewrite: \(x_{1}=-\frac{1}{4}-\frac{\sqrt{15}}{4}i,x_{2}=-\frac{1}{4}+\frac{\sqrt{15}}{4}i\) - step10: Remove the complex number(s): \(\textrm{No real solution}\) The quadratic equation \( -2x^2 - x - 2 = 0 \) has no real solutions, which means there are no x-intercepts. This indicates that the parabola does not cross the x-axis. ### Summary of Features 1. **Vertex**: \( \left(-\frac{1}{4}, -\frac{19}{8}\right) \) or \( \left(-\frac{1}{4}, -2.375\right) \) 2. **Y-intercept**: \( (0, -2) \) 3. **X-intercepts**: None (the parabola does not intersect the x-axis) If you need further analysis or specific features of the function, feel free to ask!

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

Extra Insights

Did you know that the function \( f(x) = -2x^2 - x - 2 \) is a quadratic equation? Quadratics are parabolas, and since the leading coefficient is negative (-2), this means the parabola opens downward. The vertex of this parabola will give you the maximum point, which is perfect for optimization problems! To really grasp how parabolas can be applied in real life, consider engineers designing roller coasters. They need to ensure that the curves of the tracks are safe and exhilarating! By modeling the curves with equations like this quadratic, they can calculate the highest points and slopes to create a thrilling experience while remaining structurally sound.

preguntas relacionadas

Latest Algebra Questions

¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad