Which of the following statements is true for the quadratic equation [tex]x^2-16 x+64=0[/tex]?
A. It's impossible to determine how many solutions the equation has.
B. The equation has no solutions.
C. The equation has two solutions.
D. The equation has one solution.
Answer (1)
Calculate the discriminant Δ = b 2 − 4 a c with a = 1 , b = − 16 , and c = 64 .
Find that Δ = ( − 16 ) 2 − 4 ( 1 ) ( 64 ) = 256 − 256 = 0 .
Since Δ = 0 , the equation has one real solution.
Therefore, the quadratic equation x 2 − 16 x + 64 = 0 has one solution: The equation has one solution.
Explanation
Understanding the Problem We are given the quadratic equation x 2 − 16 x + 64 = 0 . Our goal is to determine how many solutions this equation has. To do this, we will use the discriminant.
Identifying Coefficients and the Discriminant 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 = 1 , b = − 16 , and c = 64 .
Calculating the Discriminant Now, we calculate the discriminant: Δ = ( − 16 ) 2 − 4 ( 1 ) ( 64 ) = 256 − 256 = 0
Interpreting the Discriminant The discriminant tells us about the nature of the solutions:
If 0"> Δ > 0 , the equation has two distinct real solutions.
If Δ = 0 , the equation has one real solution (a repeated root).
If Δ < 0 , the equation has no real solutions.
Determining the Number of Solutions Since Δ = 0 , the quadratic equation has one real solution (a repeated root).
Final Answer Therefore, the correct statement is: The equation has one solution.
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and perimeter, and modeling the growth or decay of populations. For example, if you want to build a rectangular garden with an area of 64 square meters and you have 16 meters of fencing material, you can use a quadratic equation to find the dimensions of the garden that maximize the enclosed area. Understanding the number of solutions helps in determining if a feasible solution exists for the given constraints.
