For the quadratic expression f(x) = 5 + 20x - 4x^2 with real x, what is the maximum value of f(x)?

Introduction / Context:
This question tests understanding of quadratic functions and finding their maximum or minimum values. In aptitude exams, recognising that a quadratic with negative leading coefficient has a maximum at its vertex is very important.

Given Data / Assumptions:

• Quadratic expression: f(x) = 5 + 20x - 4x^2.
• Coefficient of x^2 is -4, which is negative.
• x can be any real number.
• We need the maximum possible value of f(x).

Concept / Approach:
A general quadratic function f(x) = ax^2 + bx + c, with a negative, opens downwards and therefore has a maximum value at its vertex. The x coordinate of the vertex is given by x = -b / (2a). The maximum value is f of that x coordinate. We can apply this directly to the given quadratic.

Step-by-Step Solution:
Step 1: Rewrite f(x) in standard form: f(x) = -4x^2 + 20x + 5.Step 2: Identify a = -4, b = 20, c = 5.Step 3: Compute the x coordinate of the vertex using x = -b / (2a).Step 4: x = -20 / (2 * -4) = -20 / -8 = 2.5.Step 5: Evaluate f(2.5) to find the maximum value.Step 6: Compute f(2.5) = -4*(2.5)^2 + 20*(2.5) + 5.Step 7: (2.5)^2 = 6.25, so -4*6.25 = -25.Step 8: 20*(2.5) = 50.Step 9: Add the terms: -25 + 50 + 5 = 30.Step 10: Therefore, the maximum value of f(x) is 30.

Verification / Alternative check:
We can complete the square: f(x) = -4x^2 + 20x + 5 = -4(x^2 - 5x) + 5. Then write x^2 - 5x as (x - 2.5)^2 - 6.25. So f(x) = -4[(x - 2.5)^2 - 6.25] + 5 = -4(x - 2.5)^2 + 25 + 5 = -4(x - 2.5)^2 + 30. The term -4(x - 2.5)^2 is always less than or equal to 0, with maximum 0 at x = 2.5, so the maximum value is 30.

Why Other Options Are Wrong:

• 1 and 5: These are too small and correspond to values of x away from the vertex.
• 25: This is close but arises from an intermediate step in completing the square, not the final maximum value.

Common Pitfalls:
Some students misuse the formula for the vertex or forget that a negative leading coefficient means a maximum, not a minimum. Others might plug random values of x instead of using the systematic vertex method, which can miss the true maximum.

Final Answer:
The maximum value of the quadratic expression is 30.