Solve by completing the square: $x^2-4 x+2=0$.
A. $2 \pm \sqrt{3}$
B. $-2 \pm \sqrt{2}$
C. $2 \pm \sqrt{2}$
D. $3 \pm \sqrt{2}$
Answer (2)
Rewrite the equation x 2 − 4 x + 2 = 0 as x 2 − 4 x = − 2 .
Complete the square by adding ( 2 − 4 ) 2 = 4 to both sides: x 2 − 4 x + 4 = − 2 + 4 , which simplifies to ( x − 2 ) 2 = 2 .
Take the square root of both sides: x − 2 = ± 2 .
Solve for x : x = 2 ± 2 . The final answer is 2 ± 2 .
Explanation
Problem Analysis We are given the quadratic equation x 2 − 4 x + 2 = 0 . Our goal is to solve for x by completing the square.
Isolating the x terms First, we rewrite the equation by moving the constant term to the right side: x 2 − 4 x = − 2
Completing the Square To complete the square, we need to add a value to both sides of the equation that will make the left side a perfect square trinomial. We take half of the coefficient of the x term, which is − 4 , and square it: ( 2 − 4 ) 2 = ( − 2 ) 2 = 4
So, we add 4 to both sides of the equation: x 2 − 4 x + 4 = − 2 + 4
Factoring Now, we can rewrite the left side as a squared term: ( x − 2 ) 2 = 2
Taking the Square Root Next, we take the square root of both sides of the equation: x − 2 = ± 2
Solving for x Finally, we solve for x by adding 2 to both sides: x = 2 ± 2
Final Answer Therefore, the solutions are x = 2 + 2 and x = 2 − 2 . Comparing with the given options, the correct answer is C.
Examples
Completing the square is a useful technique in various real-world applications. For example, engineers use it to analyze the stability of systems, economists use it to optimize cost functions, and physicists use it to solve problems in mechanics and electromagnetism. Consider an optimization problem where you want to minimize the cost function C ( x ) = x 2 − 4 x + 2 . By completing the square, you can rewrite this as C ( x ) = ( x − 2 ) 2 − 2 , which reveals that the minimum cost occurs when x = 2 , and the minimum cost is -2. This technique allows for easy identification of minimum or maximum values in quadratic functions, which is invaluable in many fields.
By completing the square for the equation x 2 − 4 x + 2 = 0 , we isolate the variable, add to create a perfect square, take the square root, and solve for x . The final solutions are 2 + 2 and 2 − 2 , which correspond to option C. Therefore, the correct choice is C: 2 ± 2 .
;
