Sure, let's go through the process of rewriting the given quadratic equation [tex]\( y = x^2 - 4x - 21 \)[/tex] in its vertex form by completing the square.
### Step-by-Step Solution:
1. Start with the given equation:
[tex]\[
y = x^2 - 4x - 21
\][/tex]
2. Isolate the [tex]\(x\)[/tex]-terms and the constant term:
[tex]\[
y = (x^2 - 4x) - 21
\][/tex]
3. Complete the square for the quadratic expression [tex]\(x^2 - 4x\)[/tex]:
- Take the coefficient of [tex]\(x\)[/tex] (which is [tex]\(-4\)[/tex]) and halve it to get [tex]\(-2\)[/tex].
- Square this result to get [tex]\(4\)[/tex].
- Add and subtract this square inside the parentheses:
[tex]\[
x^2 - 4x = (x^2 - 4x + 4) - 4
\][/tex]
4. Rewrite the equation using the completed square:
[tex]\[
y = (x^2 - 4x + 4) - 4 - 21
\][/tex]
Simplify this to:
[tex]\[
y = (x - 2)^2 - 25
\][/tex]
5. Expression in vertex form:
The vertex form of a quadratic equation is [tex]\( y = a(x - h)^2 + k \)[/tex].
Here, [tex]\(a = 1\)[/tex], [tex]\(h = 2\)[/tex], and [tex]\(k = -25\)[/tex], so:
[tex]\[
y = (x - 2)^2 - 25
\][/tex]
### Identifying the Vertex:
The vertex of the parabola [tex]\( y = (x - 2)^2 - 25 \)[/tex] is at the point [tex]\((h, k) = (2, -25)\)[/tex].
### Identifying the Minimum or Maximum Value:
For the quadratic equation in the form [tex]\( y = a(x - h)^2 + k \)[/tex]:
- If [tex]\(a > 0\)[/tex], the parabola opens upwards and has a minimum value at the vertex [tex]\( (h, k) \)[/tex].
- If [tex]\(a < 0\)[/tex], the parabola opens downwards and has a maximum value at the vertex [tex]\( (h, k) \)[/tex].
Here, [tex]\( a = 1 \)[/tex], which is positive. Therefore, the vertex [tex]\((2, -25)\)[/tex] represents the minimum point of the parabola.
So, the minimum value of the function [tex]\( y = x^2 - 4x - 21 \)[/tex] is [tex]\(-25\)[/tex].
### Conclusion:
- The vertex form of the equation [tex]\( y = x^2 - 4x - 21 \)[/tex] is [tex]\( y = (x - 2)^2 - 25 \)[/tex].
- The minimum value of the function is [tex]\(-25\)[/tex].
