To determine the correct statements about completing the square, let's analyze the equations and the choices provided.

Problem Breakdown:

1. Completing the square for $x^2 + bx$:

   • Start with $x^2 + bx$.
   • Add and subtract $\left(\frac{b}{2}\right)^2$ to complete the square: $x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2.$

2. Completing the square for $x^2 - bx$:

   • Start with $x^2 - bx$.
   • Add and subtract $\left(\frac{b}{2}\right)^2$ to complete the square: $x^2 - bx = \left(x - \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2.$

Choices Analysis:

• Choice a: $\left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2$ correctly represents completing the square for $x^2 + bx$.
• Choice b: $\left(x - \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2$ correctly represents completing the square for $x^2 - bx$.
• Choice c: Incorrect; it does not match the structure of either $x^2 + bx$ or $x^2 - bx$.
• Choice d: Incorrect; the "+" sign in $\left(\frac{b}{2}\right)^2$ does not match the negative term subtracted when completing the square.

Correct Answers:

• For $x^2 + bx$, the correct choice is a.
• For $x^2 - bx$, the correct choice is b.

Would you like a step-by-step breakdown of completing the square or further clarification?

Expanding Learning:

1. How is completing the square useful in solving quadratic equations?
2. How does completing the square relate to the quadratic formula?
3. Can completing the square work with coefficients other than 1 for $x^2$?
4. What is the geometric interpretation of completing the square?
5. How does completing the square aid in graphing parabolas?

Tip:

Always ensure the coefficient of $x^2$ is 1 before completing the square; otherwise, factor it out first!