- Identify the middle term coefficient \(\frac{1}{2}\) to determine the value of \(a\) in \((x+a)^2\).
- Recognize that \(2a = \frac{1}{2}\) implies \(a = \frac{1}{4}\).
- Confirm that \(a^2 = \frac{1}{16}\) matches the original constant term.
- Rewrite the equation as \(\left(x + \frac{1}{4}\right)^2 = \frac{4}{9}\).
- Final answer: \(\boxed{\frac{1}{4}}\).
Explanation
- Analyzing the Quadratic Expression
We are given the equation \(x^2 + \frac{1}{2}x + \frac{1}{16} = \frac{4}{9}\). The goal is to factor the expression on the left side, \(x^2 + \frac{1}{2}x + \frac{1}{16}\), into a perfect-square trinomial of the form \((x + a)^2\). We recall that a perfect-square trinomial takes the form \(x^2 + 2ax + a^2 = (x + a)^2\).
- Identifying the Constant and Factoring the Trinomial
To determine the constant \(a\), we compare our expression \(x^2 + \frac{1}{2}x + \frac{1}{16}\) to the general form \(x^2 + 2ax + a^2\). By observing the middle term, we have:
\[2a = \frac{1}{2}\]
Dividing both sides by 2 gives:
\[a = \frac{1}{2} \div 2 = \frac{1}{4}\]
Now, we must verify that this value of \(a\) satisfies the constant term in our expression:
\[a^2 = \left(\frac{1}{4}\right)^2 = \frac{1}{16}\]
Since \(\frac{1}{16}\) matches the constant term in our original expression, the trinomial is indeed a perfect square of \((x + \frac{1}{4})^2\).
- Final Factored Form
By substituting the factored form \((x + \frac{1}{4})^2\) back into our original equation, we obtain:
\[\left(x + \frac{1}{4}\right)^2 = \frac{4}{9}\]
This result provides the value to fill in the box, which is \(\frac{1}{4}\).
Examples
Perfect-square trinomials are essential in algebra for solving quadratic equations by completing the square. For example, if you are designing a rectangular garden with area \(A\) where the length and width are related to a variable \(x\), you often encounter expressions that can be simplified into squares to determine the dimensions. Mastering this factoring method allows you to simplify complex equations, making it much easier to find the values of \(x\) that satisfy your design constraints.