To determine the sign of the quadratic function [tex]\( y = x^2 - 10x + 25 \)[/tex], let's analyze its components step-by-step:
1. Quadratic Form: The given function is a quadratic function of the form [tex]\( y = ax^2 + bx + c \)[/tex]. Here:
- [tex]\( a = 1 \)[/tex] (coefficient of [tex]\( x^2 \)[/tex])
- [tex]\( b = -10 \)[/tex] (coefficient of [tex]\( x \)[/tex])
- [tex]\( c = 25 \)[/tex] (constant term)
2. Finding the Roots: First, we need to find the roots (solutions where [tex]\( y = 0 \)[/tex]). For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the roots can be found using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Let's plug in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(25)}}{2(1)}
\][/tex]
[tex]\[
x = \frac{10 \pm \sqrt{100 - 100}}{2}
\][/tex]
[tex]\[
x = \frac{10 \pm \sqrt{0}}{2}
\][/tex]
[tex]\[
x = \frac{10}{2}
\][/tex]
[tex]\[
x = 5
\][/tex]
Therefore, the function has a double root at [tex]\( x = 5 \)[/tex].
3. Vertex of the Parabola: The vertex form of a quadratic function can be useful to determine the lowest or highest point. The vertex of a parabola given by [tex]\( y = ax^2 + bx + c \)[/tex] is located at:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Substituting [tex]\( a = 1 \)[/tex] and [tex]\( b = -10 \)[/tex]:
[tex]\[
x = -\frac{-10}{2(1)} = \frac{10}{2} = 5
\][/tex]
Substituting [tex]\( x = 5 \)[/tex] back into the function to find [tex]\( y \)[/tex]:
[tex]\[
y = (5)^2 - 10(5) + 25
\][/tex]
[tex]\[
y = 25 - 50 + 25
\][/tex]
[tex]\[
y = 0
\][/tex]
So the vertex of the parabola is at [tex]\( (5, 0) \)[/tex].
4. Opening of the Parabola: Since the coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( a = 1 \)[/tex]) is positive, the parabola opens upwards.
Given that the vertex of the parabola touches the x-axis at [tex]\( (5, 0) \)[/tex] and the parabola opens upwards, the function will never take negative values. It touches the x-axis at the vertex, meaning that the function is non-negative in its entire domain.
Therefore, the sign of the quadratic function [tex]\( y = x^2 - 10x + 25 \)[/tex] is non-negative.
