Question
What is the minimum value of x^2+8x+19, and what x value does it occur at?
Answers
Answered by
GPT-4o mini
To find the minimum value of the quadratic expression \( f(x) = x^2 + 8x + 19 \), we can complete the square.
1. Start with the expression:
\[
f(x) = x^2 + 8x + 19
\]
2. To complete the square, take the coefficient of \( x \) (which is 8), divide it by 2 (giving 4), and then square it (giving 16). We will add and subtract this value inside the expression:
\[
f(x) = (x^2 + 8x + 16) + 19 - 16
\]
3. Rewrite it:
\[
f(x) = (x + 4)^2 + 3
\]
From the expression \( (x + 4)^2 + 3 \):
- The term \( (x + 4)^2 \) is a perfect square and is always non-negative. Its minimum value occurs when it equals 0, which is when \( x + 4 = 0 \) or \( x = -4 \).
- At \( x = -4 \):
\[
f(-4) = (0)^2 + 3 = 3
\]
Thus, the minimum value of \( f(x) \) is 3, which occurs at \( x = -4 \).
Hence, the minimum value is:
\[
\boxed{3}
\]
and it occurs at \( x = -4 \).
1. Start with the expression:
\[
f(x) = x^2 + 8x + 19
\]
2. To complete the square, take the coefficient of \( x \) (which is 8), divide it by 2 (giving 4), and then square it (giving 16). We will add and subtract this value inside the expression:
\[
f(x) = (x^2 + 8x + 16) + 19 - 16
\]
3. Rewrite it:
\[
f(x) = (x + 4)^2 + 3
\]
From the expression \( (x + 4)^2 + 3 \):
- The term \( (x + 4)^2 \) is a perfect square and is always non-negative. Its minimum value occurs when it equals 0, which is when \( x + 4 = 0 \) or \( x = -4 \).
- At \( x = -4 \):
\[
f(-4) = (0)^2 + 3 = 3
\]
Thus, the minimum value of \( f(x) \) is 3, which occurs at \( x = -4 \).
Hence, the minimum value is:
\[
\boxed{3}
\]
and it occurs at \( x = -4 \).