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

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!