Question
-4x^2+8x+5>=0 solve each inequality by graphing
Answers
Answered by
GPT-4o mini
To solve the inequality \(-4x^2 + 8x + 5 \geq 0\) by graphing, let's follow these steps:
1. **Rewrite the Inequality**: The inequality can be rearranged for clarity:
\[
-4x^2 + 8x + 5 \geq 0
\]
2. **Find the Roots**: To graph the quadratic, we first need to find its roots by solving the equation:
\[
-4x^2 + 8x + 5 = 0
\]
We can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = -4\), \(b = 8\), and \(c = 5\).
Calculate the discriminant:
\[
b^2 - 4ac = 8^2 - 4(-4)(5) = 64 + 80 = 144
\]
Now, calculate the roots:
\[
x = \frac{-8 \pm \sqrt{144}}{2(-4)} = \frac{-8 \pm 12}{-8}
\]
This gives us the two roots:
\[
x = \frac{-8 + 12}{-8} = \frac{4}{-8} = -\frac{1}{2} \quad \text{and} \quad x = \frac{-8 - 12}{-8} = \frac{-20}{-8} = \frac{5}{2}
\]
Thus, the roots are \(x = -\frac{1}{2}\) and \(x = \frac{5}{2}\).
3. **Graph the Quadratic**: Plot the quadratic function \(y = -4x^2 + 8x + 5\). This is a downward-opening parabola (because the coefficient of \(x^2\) is negative).
4. **Determine Where it is Above the X-axis**:
- The parabola will be above or equal to the x-axis where \(y \geq 0\).
- The x-values where the parabola crosses the x-axis are the roots we found: \(x = -\frac{1}{2}\) and \(x = \frac{5}{2}\).
- Between these two roots, the parabola will be above the x-axis.
5. **Find the Intervals**: The solution intervals are:
\[
x \in \left[-\frac{1}{2}, \frac{5}{2}\right]
\]
### Conclusion
The solution to the inequality \(-4x^2 + 8x + 5 \geq 0\) is:
\[
\boxed{\left[-\frac{1}{2}, \frac{5}{2}\right]}
\]
When you graph the function, you would see that it touches the x-axis at \(x = -\frac{1}{2}\) and \(x = \frac{5}{2}\), and is above the x-axis between these two points.
1. **Rewrite the Inequality**: The inequality can be rearranged for clarity:
\[
-4x^2 + 8x + 5 \geq 0
\]
2. **Find the Roots**: To graph the quadratic, we first need to find its roots by solving the equation:
\[
-4x^2 + 8x + 5 = 0
\]
We can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = -4\), \(b = 8\), and \(c = 5\).
Calculate the discriminant:
\[
b^2 - 4ac = 8^2 - 4(-4)(5) = 64 + 80 = 144
\]
Now, calculate the roots:
\[
x = \frac{-8 \pm \sqrt{144}}{2(-4)} = \frac{-8 \pm 12}{-8}
\]
This gives us the two roots:
\[
x = \frac{-8 + 12}{-8} = \frac{4}{-8} = -\frac{1}{2} \quad \text{and} \quad x = \frac{-8 - 12}{-8} = \frac{-20}{-8} = \frac{5}{2}
\]
Thus, the roots are \(x = -\frac{1}{2}\) and \(x = \frac{5}{2}\).
3. **Graph the Quadratic**: Plot the quadratic function \(y = -4x^2 + 8x + 5\). This is a downward-opening parabola (because the coefficient of \(x^2\) is negative).
4. **Determine Where it is Above the X-axis**:
- The parabola will be above or equal to the x-axis where \(y \geq 0\).
- The x-values where the parabola crosses the x-axis are the roots we found: \(x = -\frac{1}{2}\) and \(x = \frac{5}{2}\).
- Between these two roots, the parabola will be above the x-axis.
5. **Find the Intervals**: The solution intervals are:
\[
x \in \left[-\frac{1}{2}, \frac{5}{2}\right]
\]
### Conclusion
The solution to the inequality \(-4x^2 + 8x + 5 \geq 0\) is:
\[
\boxed{\left[-\frac{1}{2}, \frac{5}{2}\right]}
\]
When you graph the function, you would see that it touches the x-axis at \(x = -\frac{1}{2}\) and \(x = \frac{5}{2}\), and is above the x-axis between these two points.