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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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