Let's solve the given quadratic equation step by step.
### Part (a): Finding the Root(s) of the Equation [tex]\( x^2 - 2x = 0 \)[/tex]
To find the roots, we need to solve the equation [tex]\( x^2 - 2x = 0 \)[/tex].
We can factorize the quadratic equation:
[tex]\[ x^2 - 2x = x(x - 2) = 0 \][/tex]
Setting each factor equal to zero gives us:
[tex]\[ x = 0 \][/tex]
[tex]\[ x - 2 = 0 \implies x = 2 \][/tex]
So, the roots of the equation are [tex]\( x = 0 \)[/tex] and [tex]\( x = 2 \)[/tex].
Answer:
[tex]\[ \{0, 2\} \][/tex]
### Part (b): Finding the Equation of the Line of Symmetry
For a quadratic equation in the standard form [tex]\( y = ax^2 + bx + c \)[/tex], the line of symmetry (also known as the axis of symmetry) is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
In the given equation [tex]\( y = x^2 - 2x \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -2 \)[/tex]
Substituting these values into the formula:
[tex]\[ x = -\frac{-2}{2 \cdot 1} = \frac{2}{2} = 1 \][/tex]
So, the equation of the line of symmetry is:
[tex]\[ x = 1 \][/tex]
Answer:
[tex]\[ x = 1 \][/tex]
### Part (c): Finding the Coordinates of the Minimum Point
The minimum point of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] occurs at the line of symmetry. From part (b), we know the line of symmetry is [tex]\( x = 1 \)[/tex].
To find the y-coordinate of the minimum point, we substitute [tex]\( x = 1 \)[/tex] back into the original equation [tex]\( y = x^2 - 2x \)[/tex]:
[tex]\[ y = (1)^2 - 2(1) = 1 - 2 = -1 \][/tex]
So, the coordinates of the minimum point are:
[tex]\[ (1, -1) \][/tex]
Answer:
[tex]\[ (1, -1) \][/tex]
