Question

Find the product. $$\left(2 x+\frac{1}{2}\right)\left(2 x-\frac{1}{2}\right)$$

Short answer

The product of the two binomials \( \left(2 x+\frac{1}{2}\right)\left(2 x-\frac{1}{2}\right) \) is \( 4x^2 - \frac{1}{4} \).

Step-by-step solution

Expand the Binomials

The distributive property means we will distribute each term within the first parentheses across the expressions in the second parentheses: \[2x * 2x + 2x * -\frac{1}{2} + \frac{1}{2} * 2x + \frac{1}{2} * -\frac{1}{2}\]

Multiply

Perform the multiplication for each term: \[\begin{align*}4x^2 - x + x - \frac{1}{4}\end{align*}\] Note: Since \(2x * -\frac{1}{2}\) and \(\frac{1}{2} * 2x\) are the same, they can be summed up as \(1x\), hence we have \(-x + x\), which simplifies to zero.

Combine Like Terms

Now, combine like terms. In this case, there are no like terms present so the expression doesn't change: \[\begin{align*}4x^2 - \frac{1}{4}\end{align*}\]