How can you describe the solution set of the equation [tex]16 x^2-8 x+1=0[/tex]?
A. There is one real solution because the expression's factors are the same.
B. There are two different real solutions because the expression has two different factors.
C. There are no real solutions because the expression cannot be factored.
D. There are two different real solutions because the expression has three terms.
Answer (1)
The problem involves analyzing the solution set of the quadratic equation 16 x 2 − 8 x + 1 = 0 .
Calculate the discriminant: Δ = b 2 − 4 a c = ( − 8 ) 2 − 4 ( 16 ) ( 1 ) = 0 .
Since the discriminant is zero, there is one real solution.
The expression factors to ( 4 x − 1 ) 2 = 0 , confirming one repeated root.
The solution set is one real solution because the expression's factors are the same: There is one real solution because the expression’s factors are the same.
Explanation
Understanding the Problem We are given the quadratic equation 16 x 2 − 8 x + 1 = 0 and asked to describe its solution set. This involves determining the number and nature of the roots of the equation.
Calculating the Discriminant To determine the nature of the roots, we can analyze the discriminant of the quadratic equation. The discriminant, denoted as Δ , is given by the formula Δ = b 2 − 4 a c , where a , b , and c are the coefficients of the quadratic equation a x 2 + b x + c = 0 . In our case, a = 16 , b = − 8 , and c = 1 .
Evaluating the Discriminant Substituting the values of a , b , and c into the discriminant formula, we get: Δ = ( − 8 ) 2 − 4 ( 16 ) ( 1 ) = 64 − 64 = 0
Determining the Solution Set Since the discriminant Δ = 0 , the quadratic equation has exactly one real solution (also called a repeated or double root). This means the quadratic expression is a perfect square. We can verify this by factoring the quadratic expression: 16 x 2 − 8 x + 1 = ( 4 x − 1 ) 2 Setting this equal to zero, we have ( 4 x − 1 ) 2 = 0 , which gives 4 x − 1 = 0 , so x = 4 1 .
Conclusion Therefore, there is one real solution, and the expression's factors are the same.
Examples
Consider a scenario where you're designing a bridge and need to calculate the stress distribution. A quadratic equation might arise when modeling the forces acting on a particular point. If the discriminant of this equation is zero, it indicates a critical point where the stress is concentrated, and there's only one solution to the equation. This informs engineers that the design needs reinforcement at that specific point to prevent failure. Understanding the nature of quadratic solutions helps in ensuring structural integrity.
