11TH•Mathematics•Theory Of Quadratic Functions•Question 1

Step-by-Step Solution

Find the maximum or minimum value of the following quadratic functions by completing square:

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Step-by-Step Solution

Step 1: Analyze the Leading Coefficient

Comparing with $f(x) = ax^2 + bx + c$ :

a = 1, \quad b = 4, \quad c = 0

Since $a = 1 > 0$ , the parabola opens upwards ( $\cup$ ). Consequently, the function attains an absolute minimum value at its vertex.

Step 2: Complete the Square

Add and subtract $\left(\frac{b}{2}\right)^2 = \left(\frac{4}{2}\right)^2 = 2^2 = 4$ :

\begin{aligned} f(x) &= x^2 + 4x + 4 - 4 \\ &= (x + 2)^2 - 4 \end{aligned}

Step 3: Determine the Extremum Point

Because $(x + 2)^2 \ge 0$ for all real numbers $x$ , the smallest possible value of the squared term is zero:

x + 2 = 0 \implies x = -2

Step 4: Calculate the Minimum Value

Substituting $x = -2$ into the completed square form:

f(-2) = (-2 + 2)^2 - 4 = 0 - 4 = -4

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