- Identify the coefficient of the \(x\)-term, which is \(b=2\).
- Calculate the constant needed to complete the square as \(\left(\frac{b}{2}\right)^2 = \left(\frac{2}{2}\right)^2 = 1\).
- Substitute the value into the expression to get \(y = x^2 + 2x + 1 - 1 - 1\).
- Factor the perfect square trinomial to obtain the vertex form: \(y = (x+1)^2 - 2\), meaning the missing value is \(\boxed{1}\).
Explanation
- Problem Analysis
We are given the quadratic function \(y = x^2 + 2x - 1\). The objective is to convert this into vertex form, which is written as \(y = a(x-h)^2 + k\), by completing the square. The problem specifically asks us to complete the expression \(y = x^2 + 2x + \square - 1 - \square\) to form a perfect-square trinomial. To do this, we need to determine the value that completes the square.
- Identifying the Value to Complete the Square
To complete the square for a quadratic expression of the form \(x^2 + bx\), we add and subtract the value \(\left(\frac{b}{2}\right)^2\). In our case, the coefficient of the \(x\)-term is \(b = 2\). Thus, we calculate:
\[\left(\frac{b}{2}\right)^2 = \left(\frac{2}{2}\right)^2 = 1^2 = 1.\]
This is the value that must be placed in the blank squares. Adding and subtracting the same value ensures that the original expression remains unchanged.
- Completing the Square and Final Form
Now, we substitute the value 1 into the expression:
\[y = x^2 + 2x + 1 - 1 - 1.\]
Next, we identify the perfect-square trinomial \(x^2 + 2x + 1\), which factors as \((x+1)^2\). Substituting this back into the equation, we get:
\[y = (x+1)^2 - 1 - 1.\]
Finally, simplifying the constant terms \(-1 - 1 = -2\), the vertex form of the equation is:
\[y = (x+1)^2 - 2.\]
The missing value to complete the square in the provided expression is 1.
Examples
Completing the square is a powerful tool in physics and engineering. For example, if you are modeling the trajectory of a ball thrown into the air, the path is often described by a quadratic equation. By converting this equation into vertex form, you can instantly identify the vertex \((h, k)\), which represents the maximum height (k) reached by the ball at a specific time (h). This helps in predicting the peak performance or optimal range in sports like basketball or projectile motion experiments.