To find the coordinates of the minimum point of the curve [tex]\( y = x^2 - 6x + 5 \)[/tex] by completing the square, follow these steps:
1. Identify the quadratic expression: [tex]\( y = x^2 - 6x + 5 \)[/tex].
2. Rewrite the quadratic expression in the form of a perfect square trinomial:
- First, look at the coefficient of [tex]\( x \)[/tex], which is -6.
- Divide this coefficient by 2: [tex]\( \frac{-6}{2} = -3 \)[/tex].
- Square the result: [tex]\( (-3)^2 = 9 \)[/tex].
Therefore, the expression [tex]\( -6x \)[/tex] can be transformed by completing the square:
[tex]\[ y = x^2 - 6x + 9 - 9 + 5 \][/tex]
3. Group the terms to form a perfect square trinomial and simplify the constant terms:
The expression can be rewritten as:
[tex]\[ y = (x^2 - 6x + 9) - 9 + 5 \][/tex]
Here, [tex]\( x^2 - 6x + 9 \)[/tex] is a perfect square trinomial, equivalent to [tex]\( (x - 3)^2 \)[/tex].
4. Combine the constants:
[tex]\[ y = (x - 3)^2 - 4 \][/tex]
5. Identify the vertex form of the equation:
The equation [tex]\( y = (x - 3)^2 - 4 \)[/tex] is in the vertex form [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
6. Determine the vertex:
By comparing [tex]\( y = (x - 3)^2 - 4 \)[/tex] with the vertex form, we see that [tex]\( h = 3 \)[/tex] and [tex]\( k = -4 \)[/tex].
Hence, the coordinates of the minimum point (vertex) of the curve [tex]\( y = x^2 - 6x + 5 \)[/tex] are [tex]\((3, -4)\)[/tex].
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