Question

For what value of \(x\) will \(8 + (x-3)^2\) have the least value?

• A. -3
• B. 0
• C. 3
• D. 5
• E. 8

Hint:

The minimum (or maximum) value of a simple quadratic expression of the form \(a(x-h)^2 + k\) always occurs at \(x=h\). This is the vertex of the parabola. In this problem, the expression is \(1(x-3)^2 + 8\), so the minimum occurs at \(x=3\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question asks for the value of \(x\) that minimizes a given quadratic expression. The key is to understand the properties of squared terms.
Step 2: Key Formula or Approach:
The expression is \(8 + (x-3)^2\). The number 8 is a constant. To minimize the whole expression, we need to minimize the part that can vary, which is \((x-3)^2\).
The square of any real number is always greater than or equal to zero. The smallest possible value for a squared term is 0.
Step 3: Detailed Explanation:
We want to find the value of \(x\) that makes the expression \(8 + (x-3)^2\) as small as possible.
The term \((x-3)^2\) is always non-negative.
- If \((x-3)^2>0\), the total expression will be greater than 8.
- If \((x-3)^2 = 0\), the total expression will be equal to 8.
The minimum value of the expression is 8. This minimum occurs when the squared term is zero.
We need to set the squared term equal to zero and solve for \(x\):
\[ (x-3)^2 = 0 \] Take the square root of both sides:
\[ x-3 = 0 \] Add 3 to both sides:
\[ x = 3 \] When \(x=3\), the expression has its minimum value of \(8 + (3-3)^2 = 8 + 0 = 8\).
Step 4: Final Answer:
The value of \(x\) that results in the least value for the expression is 3.