Solve the quadratic equation [tex]x^2+6 x-59=0[/tex] by completing the square.
A) [tex]x=-3 \pm \sqrt{17}[/tex]
B) [tex]x=3 \pm 2 \sqrt{17}[/tex]
C) [tex]x=-3 \pm 2 \sqrt{17}[/tex]
D) [tex]x=-2 \pm 3 \sqrt{17}[/tex]

Rewrite the equation as x 2 + 6 x = 59 .
Complete the square by adding ( 6/2 ) 2 = 9 to both sides: x 2 + 6 x + 9 = 59 + 9 .
Rewrite the left side as a squared term: ( x + 3 ) 2 = 68 .
Solve for x : x = − 3 ± 2 17 ​ . The answer is x = − 3 ± 2 17 ​ ​

Explanation

Understanding the Problem We are given the quadratic equation x 2 + 6 x − 59 = 0 and asked to solve it by completing the square. Our goal is to rewrite the equation in the form ( x + a ) 2 = b , where a and b are constants. This will allow us to easily solve for x by taking the square root of both sides.

Isolating the x terms First, we rewrite the equation by adding 59 to both sides: x 2 + 6 x = 59

Completing the Square To complete the square, we need to add a constant to both sides of the equation such that the left side becomes a perfect square trinomial. The constant we need to add is ( 2 6 ​ ) 2 = 3 2 = 9 . So, we add 9 to both sides: x 2 + 6 x + 9 = 59 + 9

Rewriting as a Squared Term Now, we can rewrite the left side as a squared term: ( x + 3 ) 2 = 68

Taking the Square Root Next, we take the square root of both sides: x + 3 = ± 68 ​

Simplifying the Square Root We simplify the square root: 68 ​ = 4 ⋅ 17 ​ = 2 17 ​ . So we have: x + 3 = ± 2 17 ​

Solving for x Finally, we solve for x by subtracting 3 from both sides: x = − 3 ± 2 17 ​

Final Answer The solution to the quadratic equation is x = − 3 ± 2 17 ​ . Comparing this to the given options, we see that it matches option 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 study projectile motion. Consider a scenario where you want to find the maximum height of a projectile launched into the air. The height of the projectile can be modeled by a quadratic equation, and by completing the square, you can easily find the vertex of the parabola, which represents the maximum height. This method allows for efficient problem-solving in optimization and modeling scenarios.

Answered by GinnyAnswer | 2025-07-04

The solution to the quadratic equation x 2 + 6 x − 59 = 0 by completing the square results in x = − 3 ± 2 17 ​ . Thus, the correct option is C.
;

Answered by Anonymous | 2025-07-13