Question

Find the minimum value of the parabola y = x^2 + 4x.

Simplify your answer and write it as a proper fraction, improper fraction, or integer.

Answer 1

To find the minimum value of the parabola given by the equation \( y = x^2 + 4x \), we can use the method of completing the square.

First, we can rewrite the quadratic expression \( x^2 + 4x \).

1. Take the coefficient of \( x \), which is \( 4 \).
2. Divide it by \( 2 \) to get \( 2 \).
3. Square it to obtain \( 2^2 = 4 \).

Now, we can rewrite the expression by adding and subtracting \( 4 \):

\[
y = x^2 + 4x = (x^2 + 4x + 4) - 4 = (x + 2)^2 - 4
\]

The expression \( (x + 2)^2 \) represents a perfect square, which is always non-negative. The minimum value occurs when \( (x + 2)^2 = 0 \):

\[
(x + 2)^2 = 0 \implies x + 2 = 0 \implies x = -2
\]

Now substituting \( x = -2 \) back into the equation to find the minimum value of \( y \):

\[
y = (-2 + 2)^2 - 4 = 0 - 4 = -4
\]

Thus, the minimum value of the parabola \( y = x^2 + 4x \) is

\[
\boxed{-4}.
\]