Select the correct answer.

Solve the equation using the method of completing the square.
[tex]2 x^2+16 x-8=0[/tex]
A. [tex]x=2 \pm 4 \sqrt{5}[/tex]
B. [tex]x=4 \pm 2 \sqrt{5}[/tex]
C. [tex]x=-4 \pm 2 \sqrt{5}[/tex]
D. [tex]x=-2 \pm 4 \sqrt{5}[/tex]

Divide the equation by 2: x 2 + 8 x − 4 = 0 .
Move the constant term: x 2 + 8 x = 4 .
Complete the square: x 2 + 8 x + 16 = 20 , which simplifies to ( x + 4 ) 2 = 20 .
Solve for x : x = − 4 ± 2 5 ​ .

The correct answer is x = − 4 ± 2 5 ​ ​
Explanation

Understanding the Problem We are given the quadratic equation 2 x 2 + 16 x − 8 = 0 and asked to solve it by completing the square. This method involves manipulating the equation to create a perfect square trinomial on one side, which allows us to easily solve for x .

Simplifying the Equation First, divide the entire equation by 2 to make the coefficient of the x 2 term equal to 1: 2 2 x 2 + 16 x − 8 ​ = 2 0 ​ x 2 + 8 x − 4 = 0

Isolating the x Terms Next, move the constant term to the right side of the equation: x 2 + 8 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 = 8 , so we add ( 2 8 ​ ) 2 = 4 2 = 16 to both sides: x 2 + 8 x + 16 = 4 + 16 x 2 + 8 x + 16 = 20

Factoring the Perfect Square Now, rewrite the left side as a squared term: ( x + 4 ) 2 = 20

Taking the Square Root Take the square root of both sides of the equation: ( x + 4 ) 2 ​ = ± 20 ​ x + 4 = ± 20 ​

Simplifying the Radical Simplify the square root: 20 ​ = 4 × 5 ​ = 4 ​ × 5 ​ = 2 5 ​ x + 4 = ± 2 5 ​

Solving for x Finally, isolate x by subtracting 4 from both sides: x = − 4 ± 2 5 ​

Identifying the Correct Answer Comparing our solution x = − 4 ± 2 5 ​ with the given options, we see that it matches option C.


Examples
Completing the square is a useful technique in physics, especially when dealing with projectile motion. For example, if you want to find the maximum height of a projectile, you often end up with a quadratic equation representing the height as a function of time. By completing the square, you can rewrite the equation in vertex form, which directly gives you the maximum height and the time at which it occurs. This method allows physicists and engineers to easily determine key parameters of a projectile's trajectory, such as its range, maximum height, and time of flight, by manipulating the quadratic equations that describe its motion.

Answered by GinnyAnswer | 2025-07-03

By completing the square on the equation 2 x 2 + 16 x − 8 = 0 , we simplify it to find x = − 4 \textpm 2 5 ​ . The correct answer is option C, x = − 4 \textpm 2 5 ​ .
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Answered by Anonymous | 2025-07-04