Solve $2 x^2+20 x+8=0$ by completing the square.
A. $x=5+\sqrt{17}, x=5-\sqrt{17}$
B. $x=5+\sqrt{21}, x=5-\sqrt{21}$
C. $x=-5+\sqrt{21}, x=-5-\sqrt{21}$
D. $x=-5+\sqrt{17}, x=-5-\sqrt{17}$
Answer (1)
Divide the equation by 2: x 2 + 10 x + 4 = 0 .
Move the constant term: x 2 + 10 x = − 4 .
Complete the square: x 2 + 10 x + 25 = − 4 + 25 , which simplifies to ( x + 5 ) 2 = 21 .
Solve for x: x = − 5 ± 21 . Therefore, the solutions are x = − 5 + 21 , x = − 5 − 21 .
Explanation
Understanding the Problem We are given the quadratic equation 2 x 2 + 20 x + 8 = 0 and asked to solve it by completing the square. This method involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved.
Simplifying the Equation First, divide the entire equation by 2 to simplify it: 2 2 x 2 + 20 x + 8 = 2 0 x 2 + 10 x + 4 = 0
Isolating the x Terms Next, move the constant term to the right side of the equation: x 2 + 10 x = − 4
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. This value is ( 2 b ) 2 , where b is the coefficient of the x term. In this case, b = 10 , so we add ( 2 10 ) 2 = 5 2 = 25 to both sides: x 2 + 10 x + 25 = − 4 + 25 x 2 + 10 x + 25 = 21
Factoring the Perfect Square Trinomial Now, rewrite the left side as a squared term: ( x + 5 ) 2 = 21
Taking the Square Root Take the square root of both sides of the equation: ( x + 5 ) 2 = ± 21 x + 5 = ± 21
Solving for x Solve for x by subtracting 5 from both sides: x = − 5 ± 21
Final Answer Therefore, the solutions are x = − 5 + 21 and x = − 5 − 21 .
Examples
Completing the square is a useful technique in various real-world applications. For example, engineers use it to analyze parabolic trajectories in projectile motion, optimizing launch angles and distances. In finance, completing the square helps in maximizing profit functions or minimizing cost functions, providing insights into optimal production levels and pricing strategies. This method also finds applications in physics, such as determining the minimum potential energy in a system.
