To rewrite the quadratic function from standard form to vertex form, we need to complete the square. The given quadratic function is:
[tex]\[ f(x) = 2x^2 - 20x + 26 \][/tex]
Here are the steps to convert it to vertex form:
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the first two terms:
[tex]\[ f(x) = 2(x^2 - 10x) + 26 \][/tex]
2. Complete the square inside the parentheses:
To complete the square, take the coefficient of [tex]\( x \)[/tex] (which is -10), divide it by 2, and square the result:
[tex]\[ \left( \frac{-10}{2} \right)^2 = (-5)^2 = 25 \][/tex]
Add and subtract this square inside the parentheses:
[tex]\[ f(x) = 2(x^2 - 10x + 25 - 25) + 26 \][/tex]
Notice that we added and subtracted 25 inside the parentheses to keep the expression equivalent to the original.
3. Group the perfect square trinomial and simplify:
[tex]\[ f(x) = 2((x^2 - 10x + 25) - 25) + 26 \][/tex]
[tex]\[ f(x) = 2(x - 5)^2 - 2 \cdot 25 + 26 \][/tex]
[tex]\[ f(x) = 2(x - 5)^2 - 50 + 26 \][/tex]
4. Combine the constants:
[tex]\[ f(x) = 2(x - 5)^2 - 24 \][/tex]
So, the vertex form of the quadratic function [tex]\( f(x) = 2x^2 - 20x + 26 \)[/tex] is:
[tex]\[ f(x) = 2(x - 5)^2 - 24 \][/tex]
This shows that the vertex of the parabola is at [tex]\( (h, k) = (5, -24) \)[/tex].
