Answer :
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.
