If y = x^2 - 6x + 9, What is the Value of x? GMAT Data Sufficiency

Question: If y = x^2 - 6x + 9, what is the value of x?

(1) y = 0
(2) x + y = 3

1. Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.
2. Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: A

Solution and Explanation:
Approach Solution 1:
We can solve the problem by using the middle term split.
Given to us is an equation, y = x^2 - 6x + 9
Here y is a quadratic equation on variable x.
It has asked the value of x for the given two conditions in the questions. We have to find out if these statements are sufficient to get the answer.
As given in statement 1,
y = 0
We are given the value of y to be zero. So we’ll use this condition to get the value of x
Putting the value of y in the equation.
x^2 - 6x + 9 = 0
x^2 - (3x +3x) + 9 = 0
x^2 - 3x -3x + 9 = 0
x(x-3) -3(x-3) = 0
(x-3)^2 = 0
We got (x-3)^2 = 0
=> x = 3
Therefore this statement is sufficient to answer the problem and the answer is 3.
From the statement 2, we get,
X+y = 3
In order to find the value of x we have to replace the value of y in the equation.
Putting the value of y in the equation.
Y = x^2 - 6x + 9
=> 3 - x = x^2 - 6x + 9
=> x^2 -5x + 6 = 0
=> x^2 - (2x +3x) + 6 = 0
=> x^2 - 2x - 3x + 6x = 0
=> x(x-2) -3(x-2) = 0
=> (x-3)(x-2) = 0
The solution for this equation is x = 2,3
Since there are two values of x, this statement is not sufficient to answer the question.

Approach Solution 2:
We can solve the problem by using the quadratic formula.
Given to us, y = x^2 - 6x + 9
It has asked the value of x for the given two conditions in the questions. We have to find out if these statements are sufficient to get the answer.
As given in statement 1,
y = 0
We are given the value of y to be zero. So we’ll use this condition to get the value of x
Putting the value of y in the equation.
x^2 - 6x + 9 = 0
Using Quadratic formula,

\(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
\(x=\frac{6\pm\sqrt{(-6)^2-4.1.9}}{2}\)

x = (6\(\pm\)0)/2

x = 3

Therefore this statement is sufficient to answer the problem and the answer is 3.
From the statement 2 we get,
x+y = 3

In order to find the value of x we have to replace the value of y in the equation.
Putting the value of y in the equation.
y = x^2 - 6x + 9
=> 3 - x = x^2 - 6x + 9
=> x^2 -5x + 6 = 0

=>\(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)

\(x=\frac{5\pm\sqrt{(-5)^2-4.1.6}}{2}\)
x = (5\(\pm\)1)/2
x = \(\frac{(5-1)}{2}\) , \(\frac{(5+1)}{2}\)
x = 2,3

The solution for this equation is x = 2,3
Since there are two values of x, this statement is not sufficient to answer the question.

Approach Solution 3:
The problem statement states that y = x^2 - 6x + 9. It is asked to find the value of x.
From the equation, y = x^2 - 6x + 9, we get:
=> y= (x−3)^2

Statement 1 alone: y = 0
Therefore, we can say, y= (x−3)^2 = 0
Hence, the value of x = 3
Therefore, statement one alone is SUFFICIENT

Statement 2 alone: x + y = 3
From this statement, we cannot derive the value of x since the value of y is not known.
Therefore, statement two alone is NOT SUFFICIENT.

“If y = x^2 - 6x + 9, what is the value of x”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide 2019”. The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions are characterized by a problem statement and followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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