Solve $2 x^2-6 x+10=0$ by completing the square.
A) $x=3 / 2+i \sqrt{5}, x=3 / 2-i \sqrt{5}$
B) $x=3 / 2+1 / 2 i \sqrt{11}, x=3 / 2-1 / 2 i \sqrt{11}$
C) $x=3+i, x=3-i$
D) $x=3+i \sqrt{11}, x=3-i \sqrt{11}$
Answer (1)
Divide the equation by 2: x 2 − 3 x + 5 = 0 .
Move the constant term: x 2 − 3 x = − 5 .
Complete the square: ( x − 2 3 ) 2 = − 4 11 .
Solve for x: x = 2 3 ± 2 i 11 .
x = 2 3 + 2 1 i 11 , x = 2 3 − 2 1 i 11
Explanation
Problem Analysis We are given the quadratic equation 2 x 2 − 6 x + 10 = 0 and asked to solve it by completing the square.
Divide by 2 First, divide the entire equation by 2 to simplify it: x 2 − 3 x + 5 = 0
Isolate x terms Next, move the constant term to the right side of the equation: x 2 − 3 x = − 5
Complete the square To complete the square, we need to add ( 2 b ) 2 to both sides of the equation, where b is the coefficient of the x term. In this case, b = − 3 , so we add ( 2 − 3 ) 2 = 4 9 to both sides: x 2 − 3 x + 4 9 = − 5 + 4 9
Rewrite and simplify Now, rewrite the left side as a squared term and simplify the right side: ( x − 2 3 ) 2 = − 4 20 + 4 9 ( x − 2 3 ) 2 = − 4 11
Take the square root Take the square root of both sides: x − 2 3 = ± − 4 11 x − 2 3 = ± 2 i 11
Solve for x Finally, solve for x :
x = 2 3 ± 2 i 11
Final Answer Thus, the solutions are x = 2 3 + 2 i 11 and x = 2 3 − 2 i 11 . This corresponds to option B.
Examples
Completing the square is a useful technique in physics, especially when dealing with simple harmonic motion or analyzing projectile motion. For example, if you have an equation describing the height of a projectile as a function of time, completing the square can help you find the maximum height and the time at which it occurs. This method transforms the equation into a form that reveals key parameters, making it easier to understand and predict the behavior of physical systems. It's also used in optimization problems to find the maximum or minimum values of quadratic functions, which can represent various physical quantities.
