Bao Yu is completing the square to solve the polynomial equation:
[tex]x^2-x-3=0[/tex]
What should be her first step?
A. adding [tex]$-x$[/tex] to both sides of the equation
B. adding -3 to both sides of the equation
C. isolating the first term, [tex]$x^2$[/tex]
D. isolating the constant, -3
Answer (2)
Isolate the constant term by adding its additive inverse to both sides of the equation.
Add 3 to both sides of the equation: x 2 − x − 3 + 3 = 0 + 3 .
Simplify the equation to x 2 − x = 3 .
The first step is adding 3 to both sides of the equation. a dd in g 3 t o b o t h s i d es o f t h e e q u a t i o n
Explanation
Understanding the Problem The problem is to identify the first step in completing the square for the quadratic equation x 2 − x − 3 = 0 . Completing the square is a method used to rewrite a quadratic equation in a form that allows us to easily find the solutions.
Isolating the Constant Term To complete the square, we want to isolate the constant term on one side of the equation. This means we need to add the additive inverse of the constant term to both sides of the equation. In this case, the constant term is -3.
Adding 3 to Both Sides So, we add 3 to both sides of the equation: x 2 − x − 3 + 3 = 0 + 3 x 2 − x = 3
Conclusion Therefore, the first step in completing the square for the given equation is to add 3 to both sides of the equation.
Examples
Completing the square is a useful technique in various real-world scenarios. For instance, engineers use it to optimize designs, such as determining the dimensions of a rectangular enclosure that maximizes the enclosed area for a given perimeter. Similarly, in physics, completing the square can help find the minimum potential energy of a system, leading to stable equilibrium configurations. This method provides a structured approach to solving optimization problems across different fields.
The first step in completing the square for the equation x 2 − x − 3 = 0 is to add 3 to both sides. This isolates the variable terms on one side and prepares for the next steps of completing the square. Thus, the correct answer is option B: adding -3 to both sides of the equation.
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