To convert the quadratic equation [tex]\( y = x^2 - 14x + 52 \)[/tex] into its vertex form, we first need to identify the vertex [tex]\((h, k)\)[/tex].
The general form of a quadratic equation is:
[tex]\[ y = ax^2 + bx + c \][/tex]
The vertex form of a quadratic equation is:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola. The formula to find the vertex [tex]\( (h, k) \)[/tex] is:
[tex]\[ h = -\frac{b}{2a} \][/tex]
[tex]\[ k = c - \frac{b^2}{4a} \][/tex]
For the given equation [tex]\( y = x^2 - 14x + 52 \)[/tex]:
- The coefficient [tex]\( a \)[/tex] is 1
- The coefficient [tex]\( b \)[/tex] is -14
- The constant term [tex]\( c \)[/tex] is 52
Calculate [tex]\( h \)[/tex]:
[tex]\[ h = -\frac{b}{2a} = -\frac{-14}{2 \cdot 1} = \frac{14}{2} = 7 \][/tex]
Calculate [tex]\( k \)[/tex]:
[tex]\[ k = c - \frac{b^2}{4a} = 52 - \frac{(-14)^2}{4 \cdot 1} = 52 - \frac{196}{4} = 52 - 49 = 3 \][/tex]
Thus, the vertex [tex]\((h, k)\)[/tex] is [tex]\((7, 3)\)[/tex].
Therefore, the vertex form of the quadratic equation is:
[tex]\[ y = 1(x - 7)^2 + 3 \][/tex]
So, the correct answer is:
[tex]\[ y = (x - 7)^2 + 3 \][/tex]
