If \(x^2\) -3x=10, what is the value of x? GMAT data sufficiency

Question: If x²−3x=10, what is the value of x

1. x² − 4 = 0
2. x < 6

A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are not sufficient.

Answer: A

Solution and Explanation:

Approach Solution 1:
Given in the question a quadratic equation in the question provides us with two potential possibilities for x.
1) This quadratic equation will also provide us with two solutions. We'll determine where the two sets of answers overlap and arrive at just one solution.
Hence this is sufficient condition.
2) Here, there is no other option than to return to the initial equation and resolve it:
=> x² - 3x = 10.
=> x² - 3x - 10 = 0
=> (x-5)(x+2) = 0
X = 5, -2
This statement doesn't help us at all; it is insufficient because BOTH of these values are less than 6!
Correct option: A

Approach Solution 2:
Assumed: x² - 3x = 10
10 from either side is subtracted to yield: x² - 3x - 10 = 0
Factor: ( x + 2)(x - 5) = 0
IF x = 5 OR x = -2, then
What does x equal in terms of value?
First claim: x² - 4 = 0.
Factor: (x + 2)(x - 2) = 0
EITHER x = -2 OR x = 2 follows.
We can see that x must equal -2 when we add this to the information that either x = 5 or x = -2.
Statement 1 is sufficient because we are confident in our ability to respond to the target question.
Indicator 2: x < 6
We already know that x must either equal 5 or -2.
Statement 2 doesn't help us because 5 and -2 are both less than 6.
It MAY happen that x = 5 or it COULD happen that x = -2.
Statement 2 is not sufficient because we are unable to definitively respond to the target question.
Correct option: A

Approach Solution 3:
Modifying the initial condition and the query is the first phase of the VA (variable approach) approach. The question is then checked again.
The original condition x² - 3x = 10 is equal to x = - 2 or x = 5, as illustrated below:
X²−3x = 10
=>x²−3x−10= 0
=>(x+2)(x−5)=0
=> x = -2, 5
First condition: x² - 4 = 0
=>(x−2)(x+2)=0
=> x = −2, 2
We only have one answer because x=2 also meets the initial requirement.
Thus, the first requirement is enough.
Condition 2) Because x <6 from condition 2 and x = -2 or x=5 from the initial condition, x = -2 or x= 5 respectively.
As condition 2) is not true, we lack a singular solution.
Correct option: A

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