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

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

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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!

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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.

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