To complete the square for each quadratic expression, let's go through each step-by-step.

1. $x^2 + 3x +$

• Step 1: Take the coefficient of $x$, which is 3, divide it by 2, and square the result: $\left(\frac{3}{2}\right)^2 = \frac{9}{4}$
• Step 2: Add and subtract this value to complete the square: $x^2 + 3x + \frac{9}{4} - \frac{9}{4} = \left(x + \frac{3}{2}\right)^2 - \frac{9}{4}$
• Answer: The expression inside the square is $\left(x + \frac{3}{2}\right)^2$, and the constant term added is $\frac{9}{4}$.

2. $x^2 - 5x +$

• Step 1: Take the coefficient of $x$, which is -5, divide it by 2, and square the result: $\left(\frac{-5}{2}\right)^2 = \frac{25}{4}$
• Step 2: Add and subtract this value to complete the square: $x^2 - 5x + \frac{25}{4} - \frac{25}{4} = \left(x - \frac{5}{2}\right)^2 - \frac{25}{4}$
• Answer: The expression inside the square is $\left(x - \frac{5}{2}\right)^2$, and the constant term added is $\frac{25}{4}$.

3. $x^2 + \frac{1}{3}x +$

• Step 1: Take the coefficient of $x$, which is $\frac{1}{3}$, divide it by 2, and square the result: $\left(\frac{1}{6}\right)^2 = \frac{1}{36}$
• Step 2: Add and subtract this value to complete the square: $x^2 + \frac{1}{3}x + \frac{1}{36} - \frac{1}{36} = \left(x + \frac{1}{6}\right)^2 - \frac{1}{36}$
• Answer: The expression inside the square is $\left(x + \frac{1}{6}\right)^2$, and the constant term added is $\frac{1}{36}$.

These answers are the values that should fill in the blanks on the provided worksheet.

Would you like more details on any of these steps or have any other questions?

Relative Questions:

1. How do you solve quadratic equations using the method of completing the square?
2. What is the significance of the vertex form of a quadratic function?
3. Can completing the square be used to derive the quadratic formula?
4. How do you handle completing the square when the leading coefficient is not 1?
5. What are the applications of completing the square in real-world problems?

Tip:

When completing the square, always remember to balance the equation by adding the same value on both sides if you are solving an equation.