Find the minimum value of f(x) = x^2 + 6x + 10. (2020)

Find the minimum value of f(x) = x^2 + 6x + 10. (2020)

Step 1: Identify the function we need to analyze, which is f(x) = x^2 + 6x + 10.
Step 2: Recognize that this is a quadratic function in the form of ax^2 + bx + c, where a = 1, b = 6, and c = 10.
Step 3: To find the minimum value of a quadratic function, use the formula x = -b/(2a).
Step 4: Substitute the values of a and b into the formula: x = -6/(2*1).
Step 5: Calculate -6/(2*1) which simplifies to -6/2 = -3.
Step 6: Now, substitute x = -3 back into the function to find the minimum value: f(-3) = (-3)^2 + 6(-3) + 10.
Step 7: Calculate (-3)^2 which is 9, then calculate 6*(-3) which is -18.
Step 8: Combine these results: f(-3) = 9 - 18 + 10.
Step 9: Simplify the expression: 9 - 18 = -9, and -9 + 10 = 1.
Step 10: Therefore, the minimum value of f(x) is 1.

Quadratic Functions – Understanding the properties of quadratic functions, including how to find their minimum or maximum values using the vertex formula.
Vertex Formula – Using the formula x = -b/(2a) to find the x-coordinate of the vertex of a parabola represented by a quadratic function.
Function Evaluation – Calculating the function value at a specific point to determine the minimum value of the function.