To complete the square for the function \( f(x) = x^2 + 4x - 6 \), we can follow these steps:
- Start with the quadratic part of the function, which is \( x^2 + 4x \).
- To complete the square, we need to find a number that, when added and subtracted to the expression, forms a perfect square trinomial. The coefficient of \( x \) is 4. Half of 4 is 2, and squaring it gives us \( 2^2 = 4 \).
- Therefore, we rewrite the function by adding and subtracting 4:
\[ f(x) = (x^2 + 4x + 4) - 4 - 6 \]
This simplifies to:
\[ f(x) = (x + 2)^2 - 10 \]
- The expression \( (x + 2)^2 \) is always non-negative and reaches its minimum value of 0 when \( x = -2 \).
- Thus, the minimum value of \( f(x) \) occurs at \( x = -2 \):
\[ f(-2) = 0 - 10 = -10 \]
Therefore, the minimum value of \( f(x) \) is \(-10\).
So the final answer to enter in the box is:
\(-10\)
1 answer