Question
Complete the square and find the vertex form of the quadratic function.
f(x)= x^2 -2x+5
f(x)= x^2 -2x+5
Answers
Answered by
GPT 3.5
To complete the square, we need to take the coefficient of the x term, divide it by 2, and then square the result. For the quadratic function f(x) = x^2 - 2x + 5:
1. Take half of the coefficient of the x term: -2/2 = -1
2. Square the result: (-1)^2 = 1
Now let's rewrite the quadratic function by adding and subtracting 1 inside the parentheses:
f(x) = x^2 - 2x + 1 - 1 + 5
Next, let's group the square term with the constant term:
f(x) = (x^2 - 2x + 1) - 1 + 5
The grouped terms x^2 - 2x + 1 can be written as a square of a binomial:
f(x) = (x - 1)^2 - 1 + 5
Simplifying further:
f(x) = (x - 1)^2 + 4
Therefore, the vertex form of the quadratic function is f(x) = (x - 1)^2 + 4. The vertex of the parabola is (1, 4).
1. Take half of the coefficient of the x term: -2/2 = -1
2. Square the result: (-1)^2 = 1
Now let's rewrite the quadratic function by adding and subtracting 1 inside the parentheses:
f(x) = x^2 - 2x + 1 - 1 + 5
Next, let's group the square term with the constant term:
f(x) = (x^2 - 2x + 1) - 1 + 5
The grouped terms x^2 - 2x + 1 can be written as a square of a binomial:
f(x) = (x - 1)^2 - 1 + 5
Simplifying further:
f(x) = (x - 1)^2 + 4
Therefore, the vertex form of the quadratic function is f(x) = (x - 1)^2 + 4. The vertex of the parabola is (1, 4).
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