To determine the value of [tex]\( x \)[/tex] at which [tex]\( y \)[/tex] reaches its minimum for the quadratic equation [tex]\( y = x^2 - 14x + 22 \)[/tex], we can follow these steps:
1. Identify the Standard Form:
The given quadratic equation is in the form [tex]\( y = ax^2 + bx + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -14 \)[/tex], and [tex]\( c = 22 \)[/tex].
2. Formula for the Vertex's x-coordinate:
The x-coordinate of the vertex of a parabola given by the quadratic equation can be found using the formula:
[tex]\[
x_{\text{vertex}} = -\frac{b}{2a}
\][/tex]
This formula gives the x-coordinate at which the minimum (or maximum) value of [tex]\( y \)[/tex] occurs for a parabola that opens upwards (or downwards).
3. Substitute the Values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
x_{\text{vertex}} = -\frac{-14}{2 \cdot 1} = \frac{14}{2} = 7
\][/tex]
4. Verify by Calculating the Minimum Value [tex]\( y \)[/tex]:
To find the minimum value of [tex]\( y \)[/tex], substitute [tex]\( x = 7 \)[/tex] back into the original equation:
[tex]\[
y_{\text{minimum}} = (7)^2 - 14 \cdot 7 + 22
= 49 - 98 + 22
= -27
\][/tex]
The minimum value of [tex]\( y \)[/tex] is [tex]\(-27\)[/tex] at [tex]\( x = 7 \)[/tex].
Therefore, [tex]\( y \)[/tex] reaches its minimum value when [tex]\( x = 7 \)[/tex].
