What is the solution of the equation [tex]x^2-2 x+4=0[/tex]?
Answer (1)
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c , where a = 1 , b = − 2 , and c = 4 .
Substitute the values: x = 2 ( 1 ) 2 ± ( − 2 ) 2 − 4 ( 1 ) ( 4 ) = 2 2 ± − 12 .
Simplify the square root: − 12 = 2 i 3 .
Obtain the solutions: x = 1 ± i 3 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 − 2 x + 4 = 0 . Our goal is to find the solution(s) for x . We can use the quadratic formula to solve for x .
Applying the Quadratic Formula The quadratic formula is given by x = 2 a − b ± b 2 − 4 a c , where a = 1 , b = − 2 , and c = 4 . Substituting these values into the formula, we get: x = 2 ( 1 ) − ( − 2 ) ± ( − 2 ) 2 − 4 ( 1 ) ( 4 )
Simplifying the Expression Now, let's simplify the expression: x = 2 2 ± 4 − 16 = 2 2 ± − 12
Dealing with the Imaginary Term We can rewrite the square root of -12 as − 12 = 12 i = 4 ⋅ 3 i = 2 i 3 . Substituting this back into the expression for x , we have: x = 2 2 ± 2 i 3
Finding the Solutions Finally, we divide both terms in the numerator by 2: x = 1 ± i 3 Thus, the solutions are x = 1 + i 3 and x = 1 − i 3 .
Final Answer The solution to the equation x 2 − 2 x + 4 = 0 is x = 1 ± i 3 .
Examples
Quadratic equations appear in various fields, such as physics, engineering, and economics. For example, in physics, the trajectory of a projectile under constant gravitational acceleration is described by a quadratic equation. Solving such equations helps determine the range, maximum height, and time of flight of the projectile. Understanding quadratic equations is fundamental for solving real-world problems involving parabolic motion and optimization.
