What is the solution to [tex]$5 x^2-2 x+5=0$[/tex]?
A. [tex]$x=\frac{1 \pm 2 i \sqrt{6}}{5}$[/tex]
B. [tex]$x=\frac{-2 \pm 2 i \sqrt{6}}{10}$[/tex]
C. [tex]$x=\frac{2 \pm 2 i \sqrt{6}}{5}$[/tex]
D. [tex]$x=\frac{1 \pm 2 i \sqrt{6}}{10}$[/tex]
E. [tex]$x=\frac{1 \pm 2 \sqrt{6}}{5}$[/tex]
Answer (1)
Apply the quadratic formula x = 2 a − b ± b 2 − 4 a c with a = 5 , b = − 2 , and c = 5 .
Simplify the expression to x = 10 2 ± − 96 .
Rewrite − 96 as 4 i 6 .
Simplify the final expression to obtain the solutions: x = 5 1 ± 2 i 6 .
Explanation
Problem Analysis We are given the quadratic equation 5 x 2 − 2 x + 5 = 0 . Our goal is to find the solutions for x . We can use the quadratic formula to solve this equation.
Quadratic Formula The quadratic formula is given by x = 2 a − b ± 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 = 5 , b = − 2 , and c = 5 .
Substitution Substitute the values of a , b , and c into the quadratic formula: x = 2 ( 5 ) − ( − 2 ) ± ( − 2 ) 2 − 4 ( 5 ) ( 5 )
Simplification Simplify the expression: x = 10 2 ± 4 − 100 = 10 2 ± − 96
Simplifying the Square Root Since the discriminant (the value inside the square root) is negative, we will have complex solutions. We can simplify − 96 as follows: − 96 = 16 ⋅ 6 ⋅ − 1 = 16 ⋅ 6 ⋅ − 1 = 4 i 6
Substituting Back Substitute this back into the expression for x : x = 10 2 ± 4 i 6
Final Simplification Now, simplify the fraction by dividing both the numerator and the denominator by 2: x = 5 1 ± 2 i 6
Conclusion Therefore, the solutions to the quadratic equation 5 x 2 − 2 x + 5 = 0 are x = 5 1 + 2 i 6 and x = 5 1 − 2 i 6 .
Examples
Quadratic equations are not just abstract math; they appear in various real-world applications. For example, when designing a parabolic mirror for a telescope, engineers use quadratic equations to determine the precise shape needed to focus light correctly. Similarly, in physics, projectile motion, like the trajectory of a ball thrown in the air, can be modeled using quadratic equations, helping to predict its range and maximum height. Understanding quadratic equations is essential for solving problems related to optimization, engineering, and physics.
