Question

Complete the square to determine the minimum or maximum value of the function defined by the expression.

x2 − 10x + 15
A) maximum value at −10
B) minimum value at −10
C) maximum value at −15
D) minimum value at −15

Answer 1

Option B) minimum value at −10

Explanation:

we have

`f(x)=x^(2) -10x+15`

This function represent a vertical parabola open upward (because the leading coefficient is positive)

The vertex represent a minimum

Group terms that contain the same variable, and move the constant to the opposite side of the equation

`f(x)-15=x^(2) -10x`

Divide the coefficient of term x by 2

10/2=5

squared the term and add to the right side of equation

`f(x)-15=(x^(2) -10x+5^2)`

Remember to balance the equation by adding the same constants to the other side

`f(x)-15+5^2=(x^(2) -10x+5^2)`

`f(x)+10=(x^(2) -10x+25)`

rewrite as perfect squares

`f(x)+10=(x-5)^(2)`

`f(x)=(x-5)^(2)-10` ----> function in vertex form

The vertex of the quadratic function is the point (5,-10)

therefore

The minimum value of the function is -10