Choose the equation that represents the solutions of $0=0.25 x^2-8 x$.
A. $x=\frac{8 \pm \sqrt{(-8)^2-(4)(0.25)(0)}}{2(0.25)}$
B. $x=\frac{-8 \pm \sqrt{(-8)^2-(4)(0.25)(0)}}{2(0.25)}$
C. $x=\frac{-0.25 \pm \sqrt{(0.25)^2-(4)(1)(-8)}}{2(1)}$
Answer (1)
Identify the coefficients: a = 0.25 , b = − 8 , and c = 0 .
Substitute the coefficients into the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute: x = 2 ( 0.25 ) − ( − 8 ) ± ( − 8 ) 2 − 4 ( 0.25 ) ( 0 ) .
The correct equation is: x = 2 ( 0.25 ) 8 ± ( − 8 ) 2 − ( 4 ) ( 0.25 ) ( 0 ) .
Explanation
Understanding the Problem We are given the quadratic equation 0 = 0.25 x 2 − 8 x and asked to choose the correct equation that represents its solutions using the quadratic formula.
Recalling the Quadratic Formula The quadratic formula is given by x = 2 a − b ± b 2 − 4 a c for a quadratic equation in the form a x 2 + b x + c = 0 .
Identifying Coefficients In our equation, 0.25 x 2 − 8 x = 0 , we can identify the coefficients as a = 0.25 , b = − 8 , and c = 0 .
Substituting into the Formula Now, we substitute these values into the quadratic formula: x = 2 ( 0.25 ) − ( − 8 ) ± ( − 8 ) 2 − 4 ( 0.25 ) ( 0 )
Simplifying the Expression Simplifying the expression, we get: x = 0.5 8 ± 64 − 0 x = 0.5 8 ± 64
Choosing the Correct Option Comparing this with the given options, we see that the correct equation is: x = 2 ( 0.25 ) 8 ± ( − 8 ) 2 − ( 4 ) ( 0.25 ) ( 0 )
Examples
The quadratic formula is a fundamental tool in algebra, used to solve equations that model various real-world scenarios. For example, if you're designing a bridge and need to calculate the trajectory of a cable under load, you might use a quadratic equation. Similarly, in physics, calculating the height of a projectile over time involves solving a quadratic equation. Understanding how to correctly substitute values into the quadratic formula is crucial for accurate problem-solving in these fields.
