Question

complete the square to find the minimum value of f(x) = x² + 4x - 6. enter the answer in the box. the minimum value of f(x) is box.

Explanation:

Step1: Recall completing the square formula

For a quadratic function \( f(x) = ax^2 + bx + c \), completing the square involves rewriting it in the form \( f(x)=a(x - h)^2 + k \), where \( h=-\frac{b}{2a} \) and \( k \) is the constant term after completing the square. For \( f(x)=x^2 + 4x - 6 \), \( a = 1 \), \( b = 4 \), \( c=-6 \). We take the terms with \( x \): \( x^2+4x \). To complete the square, we add and subtract \( (\frac{b}{2})^2 \). Here, \( \frac{b}{2}=\frac{4}{2} = 2 \), so \( (\frac{b}{2})^2=4 \).

Step2: Rewrite the function

\( f(x)=x^2 + 4x - 6=(x^2 + 4x + 4)-4 - 6 \). The \( x^2 + 4x + 4 \) is a perfect square, \( (x + 2)^2 \). So \( f(x)=(x + 2)^2-10 \).

Step3: Find the minimum value

Since \( (x + 2)^2\geq0 \) for all real numbers \( x \) (because the square of any real number is non - negative), the minimum value of \( (x + 2)^2 \) is \( 0 \). When \( (x + 2)^2 = 0 \) (i.e., \( x=-2 \)), \( f(x)=(0)-10=-10 \).

Answer:

\(-10\)