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) When a≠0, how many solutions does the equation

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by Max@Math Revolution » Tue Jan 23, 2018 12:05 am
[GMAT math practice question]

$$When\ \ a\ne0,\ how\ many\ solutions\ does\ the\ equation\ a\left(x+b\right)^2+c=0\ have?$$

$$\left(1\right)\ bc=0$$
$$\left(2\right)\ |b|+|c|=0$$
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Source: — Data Sufficiency |
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by Max@Math Revolution » Thu Jan 25, 2018 2:55 am
=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question, and then recheck the question.

There are three cases to consider:
Case 1: a > 0, c > 0 or a < 0, c < 0
The equation has no roots.

Case 2: c = 0.
The equation has only one root.

Case 3: a > 0, c < 0 or a < 0, c > 0
The equation has two roots.

Condition 1):
If bc = 0, then when
a = 1, b = 0, c = -1, the equation has two roots, and when
a = 1, b = 0, c = 0, the equation has one root.
As the question does not have a unique answer, condition 1) is not sufficient.

Condition 2)
|b| + |c| = 0 ⇔ b = c = 0.
Since c = 0, the equation has only one root.
Condition 2) is sufficient.

Therefore, the answer is B.

Answer: B
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by DrMaths » Thu Jan 25, 2018 6:51 am
Could someone please post the question again? I can only see the equations in code, not in a readable format.

Many thanks.
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