What is the solution to $5 x^2-2 x+5=0$ ?
A. $x=\frac{1 \pm 2 i \sqrt{6}}{5}$
B. $x=\frac{-2 \pm 2 i \sqrt{6}}{10}$
C. $x=\frac{2 \pm 2 i \sqrt{6}}{5}$
D. $x=\frac{1 \pm 2 i \sqrt{6}}{10}$
E. $x=\frac{1 \pm 2 \sqrt{6}}{5}$
Answer (1)
Apply the quadratic formula x = 2 a − b ± b 2 − 4 a c with a = 5 , b = − 2 , and c = 5 .
Substitute the values into the formula: x = 10 2 ± 4 − 100 .
Simplify the expression: x = 10 2 ± i 96 = 10 2 ± 4 i 6 .
Divide by 2 to get the final answer: x = 5 1 ± 2 i 6 .
Explanation
Understanding the Problem and the Quadratic Formula 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. 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 .
Substituting Values into the Quadratic Formula Now, we substitute the values of a , b , and c into the quadratic formula: x = 2 ( 5 ) − ( − 2 ) ± ( − 2 ) 2 − 4 ( 5 ) ( 5 ) Simplify the expression: x = 10 2 ± 4 − 100 x = 10 2 ± − 96
Simplifying the Expression Since we have a negative number inside the square root, we will have complex solutions. We can rewrite the square root of the negative number using the imaginary unit i , where i = − 1 : x = 10 2 ± i 96 Now, we simplify the radical: 96 = 16 ⋅ 6 = 4 6 Substitute this back into the expression for x : x = 10 2 ± 4 i 6
Final Simplification and Solutions Finally, we divide both the numerator and the denominator by 2: x = 5 1 ± 2 i 6 So the solutions are: x = 5 1 + 2 i 6 , x = 5 1 − 2 i 6
The Final Answer The solution to the quadratic equation 5 x 2 − 2 x + 5 = 0 is: x = 5 1 ± 2 i 6 Therefore, the correct answer is A.
Examples
Quadratic equations are not just abstract math; they appear in various real-world applications. For instance, when designing a parabolic mirror for a telescope, engineers use quadratic equations to ensure the mirror focuses light correctly. Similarly, in finance, quadratic equations can model investment growth or calculate break-even points. Understanding how to solve these equations allows for precise calculations and informed decision-making in these fields.
