Multiple choice

The minimum value of the quadratic expression ${x}^{2}+2{b}{x}+{c}$ is

  1. ${c}{b}^{2}$
  2. ${c}^{2}{b}$
  3. ${c}+{b}^{2}$
  4. ${c}-{b}^{2}$
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D Correct answer
Explanation

The expression x^2 + 2bx + c is a parabola opening upwards. The minimum occurs at x = -b. Substituting x = -b: (-b)^2 + 2b(-b) + c = b^2 - 2b^2 + c = c - b^2.

AI explanation

The minimum value of a quadratic expression ax^2 + bx + c is found using the formula c - b^2/4a. Here, a = 1, b = 2b, and c = c. Substituting these values into the formula gives c - (2b)^2 / 4. This simplifies to c - 4b^2 / 4, resulting in a minimum value of c - b^2.

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