Question

how can you describe the solution set of the equation 16x² - 8x + 1 = 0?
○ there are two different real solutions because the expression has two different factors.
○ there are two different real solutions because the expression has three terms.
○ there is one real solution because the expression’s factors are the same.
○ there are no real solutions because the expression cannot be factored.
done

Explanation:

Step1: Factor the quadratic equation

The quadratic equation is \(16x^2 - 8x + 1 = 0\). We can factor it as a perfect square trinomial. Notice that \(16x^2=(4x)^2\), \(1 = 1^2\), and \(-8x=-2\times4x\times1\). So, it factors to \((4x - 1)^2=0\).

Step2: Analyze the solution

Since \((4x - 1)^2=0\), we can set \(4x - 1 = 0\) (because a square of a number is zero only when the number itself is zero). Solving \(4x - 1 = 0\) gives \(x=\frac{1}{4}\). But since the factor \((4x - 1)\) is repeated (the square means the same factor is multiplied by itself), there is only one distinct real solution (even though the root is repeated, it's still one real solution in terms of distinct values, and the reason is that the factors are the same).

Answer:

There is one real solution because the expression’s factors are the same.