Discover how IDNLearn.com can help you find the answers you need quickly and easily. Ask your questions and receive accurate, in-depth answers from our knowledgeable community members.
Sagot :
To determine the types of roots and the positions of the quadratic function \( f(x) = x(5 - 2x) \) with respect to the x-axis, follow these steps:
1. **Rewrite the function in standard quadratic form**:
\[ f(x) = x(5 - 2x) \]
\[ f(x) = 5x - 2x^2 \]
Rewriting it in standard form \( ax^2 + bx + c \):
\[ f(x) = -2x^2 + 5x \]
Where:
\[ a = -2, b = 5, c = 0 \]
2. **Calculate the discriminant** to determine the types of roots:
The discriminant \(\Delta\) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ \Delta = b^2 - 4ac \]
Plug in the values:
\[ \Delta = 5^2 - 4(-2)(0) \]
\[ \Delta = 25 - 0 \]
\[ \Delta = 25 \]
Since \(\Delta > 0\), the quadratic function has two distinct real roots.
3. **Determine the roots**:
Use the quadratic formula \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \):
\[ x = \frac{-5 \pm \sqrt{25}}{2(-2)} \]
\[ x = \frac{-5 \pm 5}{-4} \]
Calculate the roots:
\[ x_1 = \frac{-5 + 5}{-4} = 0 \]
\[ x_2 = \frac{-5 - 5}{-4} = \frac{-10}{-4} = \frac{10}{4} = 2.5 \]
So, the roots are \(x_1 = 0\) and \(x_2 = 2.5\).
4. **Graph the function** to understand its position with respect to the x-axis:
The quadratic function \( f(x) = -2x^2 + 5x \) is a downward-facing parabola (since \(a < 0\)).
- The roots \(x = 0\) and \(x = 2.5\) are the points where the parabola intersects the x-axis.
- Since it is a downward-facing parabola, it opens downwards, thus the parabola is above the x-axis between the roots \(0\) and \(2.5\) and below the x-axis outside these roots.
In summary:
- The quadratic function \( f(x) = -2x^2 + 5x \) has two distinct real roots at \(x = 0\) and \(x = 2.5\).
- The parabola intersects the x-axis at these points and is above the x-axis between these points and below outside these points.
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your search for solutions ends here at IDNLearn.com. Thank you for visiting, and come back soon for more helpful information.
