- Calculate the discriminant of \(16x^2 - 8x + 1 = 0\), which is \(D = (-8)^2 - 4(16)(1) = 0\).
- Recognize that a discriminant of zero indicates a single repeated real root.
- Factor the expression as \((4x - 1)^2 = 0\), showing that the factors are identical.
- Conclude that there is one real solution because the expression's factors are the same.
- Final Answer: \(\boxed{\text{There is one real solution because the expression's factors are the same.}}\)
Explanation
- Analyzing the Discriminant
The given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\), specifically \(16x^2 - 8x + 1 = 0\). We identify the coefficients as \(a = 16\), \(b = -8\), and \(c = 1\). To understand the nature of the solutions, we calculate the discriminant, defined as \(D = b^2 - 4ac\). Performing the calculation, we get \(D = (-8)^2 - 4 \times 16 \times 1 = 64 - 64 = 0\). A discriminant of \(0\) indicates that the equation has exactly one real solution with multiplicity \(2\).
- Factoring the Expression
Next, we examine the quadratic expression \(16x^2 - 8x + 1\) to determine if it can be factored. We observe that it is a perfect square trinomial because it fits the pattern \((mx - n)^2 = m^2x^2 - 2mnx + n^2\). Comparing \(16x^2 - 8x + 1\) to this pattern, we have \((4x)^2 - 2(4x)(1) + 1^2\), which simplifies to \((4x - 1)^2 = 0\). This confirms that the expression is the square of a single linear factor. Since the factors are identical, setting each factor to zero yields the same result: \(4x - 1 = 0\), which results in \(x = \frac{1}{4}\). Therefore, there is only one distinct real solution.
- Conclusion
Based on our analysis of the discriminant and the factorization of the expression, we conclude that the equation has exactly one real solution because the factors are identical. Among the given choices, the correct description is: "There is one real solution because the expression's factors are the same." Therefore, the final answer is that there is one real solution.
Examples
Quadratic equations with a single repeated root often appear in real-world engineering and physics problems, such as determining the critical point in a stable mechanical system where an object is just balanced. For instance, if you are modeling the trajectory of a ball that just touches a surface, the point of contact occurs at the repeated root of the quadratic equation describing the ball's position relative to the surface. Understanding that these equations result in a single solution helps in identifying precise points of tangency or equilibrium in various physical designs and optimization tasks.