To write the quadratic function [tex]\( y = x^2 + 6x + 10 \)[/tex] in vertex form, we need to find its vertex [tex]\((h, k)\)[/tex]. The vertex form of a quadratic function is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
The general form of a quadratic equation is:
[tex]\[ y = ax^2 + bx + c \][/tex]
In our equation, we have [tex]\(a = 1\)[/tex], [tex]\(b = 6\)[/tex], and [tex]\(c = 10\)[/tex]. We can determine the vertex [tex]\((h, k)\)[/tex] using these formulas:
1. Finding [tex]\(h\)[/tex]:
The formula to find the x-coordinate of the vertex (h) is:
[tex]\[ h = -\frac{b}{2a} \][/tex]
2. Calculating [tex]\(h\)[/tex]:
[tex]\[ h = -\frac{6}{2(1)} = -\frac{6}{2} = -3 \][/tex]
3. Finding [tex]\(k\)[/tex]:
To find the y-coordinate of the vertex (k), substitute [tex]\(h = -3\)[/tex] back into the original quadratic equation:
[tex]\[ y = 1(-3)^2 + 6(-3) + 10 \][/tex]
[tex]\[ k = 1(9) + (-18) + 10 \][/tex]
[tex]\[ k = 9 - 18 + 10 \][/tex]
[tex]\[ k = 1 \][/tex]
Therefore, the vertex [tex]\((h, k)\)[/tex] is [tex]\((-3, 1)\)[/tex].
Now, substituting [tex]\(h\)[/tex] and [tex]\(k\)[/tex] back into the vertex form, we get:
[tex]\[ y = (x + 3)^2 + 1 \][/tex]
So, the quadratic function [tex]\( y = x^2 + 6x + 10 \)[/tex] in vertex form is:
[tex]\[ y = (x + 3)^2 + 1 \][/tex]
The values to complete the equation are:
[tex]\[ y = (x + \square)^2 + \square \][/tex]
[tex]\[ y = (x + 3)^2 + 1 \][/tex]
