Let's analyze the quadratic equation [tex]\( x^2 - 6x + 2 = 0 \)[/tex] step by step.
1. Determine the nature of the extreme value:
- The quadratic equation is in the form [tex]\( ax^2 + bx + c \)[/tex].
- Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = 2 \)[/tex].
- Since [tex]\( a > 0 \)[/tex] (specifically [tex]\( a = 1 \)[/tex]), the parabola opens upwards. Therefore, the graph of the quadratic equation has a minimum value, not a maximum.
Correct statement(s):
* - "The graph of the quadratic equation has a minimum value."
2. Calculate the vertex (extreme value) of the parabola:
- The x-coordinate of the vertex is given by [tex]\( x = -\frac{b}{2a} \)[/tex].
- Plugging in the values: [tex]\( x = -\frac{-6}{2 \cdot 1} = \frac{6}{2} = 3 \)[/tex].
- To find the y-coordinate, substitute [tex]\( x = 3 \)[/tex] back into the equation:
[tex]\[
y = a \cdot (3)^2 + b \cdot 3 + c = 1 \cdot 9 - 6 \cdot 3 + 2 = 9 - 18 + 2 = -7
\][/tex]
- So, the vertex (extreme value) is [tex]\( (3, -7) \)[/tex].
Correct statement(s):
* - "The extreme value is at the point [tex]\( (3, -7) \)[/tex]."
3. Determine the solutions of the quadratic equation:
- The solutions of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] are found using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
- Calculate the discriminant [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[
\text{Discriminant} = (-6)^2 - 4 \cdot 1 \cdot 2 = 36 - 8 = 28
\][/tex]
- The solutions, using the discriminant:
[tex]\[
x = \frac{6 \pm \sqrt{28}}{2}
\][/tex]
- Simplifying further:
[tex]\[
x = \frac{6 \pm 2\sqrt{7}}{2} = 3 \pm \sqrt{7}
\][/tex]
- Therefore, the solutions of the quadratic equation are [tex]\( x = 3 \pm \sqrt{7} \)[/tex].
Correct statement(s):
- "The solutions are [tex]\( x = 3 \pm \sqrt{7} \)[/tex]."
Incorrect statements:
- "The graph of the quadratic equation has a maximum value." (Incorrect, it has a minimum value.)
- "The extreme value is at the point [tex]\( (7, -3) \)[/tex]." (Incorrect point.)
- "The solutions are [tex]\( x = -3 \pm \sqrt{7} \)[/tex]." (Incorrect solutions.)
Summary of correct statements:
"The graph of the quadratic equation has a minimum value."
"The extreme value is at the point [tex]\( (3, -7) \)[/tex]."
* "The solutions are [tex]\( x = 3 \pm \sqrt{7} \)[/tex]."
