Question

bao yu is completing the square to solve the polynomial equation.\\(x^2 - x - 3 = 0\\)\
what should be her first step?\
adding \\(-3\\) to both sides of the equation\
isolating the first term, \\(x^2\\)\
adding \\(-x\\) to both sides of the equation

Explanation:

Step1: Recall completing the square rules

To complete the square, constant terms are moved to the opposite side of the variable terms first.

Step2: Analyze the given equation

For $x^2 - x - 3 = 0$, we need to isolate the variable terms. This requires adding 3 to both sides (which is the inverse of subtracting 3). We can eliminate incorrect options:

• Adding $-3$ would make the constant term $-6$, which is wrong.
• Isolating only $x^2$ leaves $-x$ on the wrong side, which is not the first step.
• Adding $-x$ would make the $x$-term $-2x$, which is incorrect.

The correct first step is to move the constant $-3$ to the right side by adding 3 to both sides, which aligns with isolating the variable terms.

Step1: Recall completing the square process

First, isolate all variable terms on one side, constants on the other.

Step2: Apply to the given equation

For $x^2 - x - 3 = 0$, add 3 to both sides:
$x^2 - x - 3 + 3 = 0 + 3$
$x^2 - x = 3$

Step3: Evaluate provided options

• Adding $-3$: $x^2 - x -6=0$, incorrect.
• Isolating $x^2$: $x^2 = x + 3$, not standard first step.
• Adding $-x$: $x^2 -2x -3=0$, incorrect.

Answer:

(Note: The correct option matching "adding 3 to both sides of the equation" is not listed as written, but re-evaluating the options, the intended correct action is to isolate variable terms. The only valid first step logic is moving the constant: the correct action equivalent is that we need to add 3 to both sides, but since the options provided have a typo, the logical correct approach is that the first step is to get the constant on the other side. However, based on the options, the correct intended answer is that none of the written options are perfect, but the standard first step is to add 3 to both sides. But since the options are as given, the correct reasoning leads to:
The correct first step is to add 3 to both sides, which is the inverse of the incorrect "adding -3" option. The valid first step for completing the square here is to isolate the variable terms by moving the constant to the right, so the correct action is adding 3 to both sides. Since this is not listed, but if we assume a typo, the intended correct option would be the action that moves the constant, so the answer is that the first step is to add 3 to both sides, and the given options have an error, but the logical correct step is: