Which of the Following Equations has A Root in Common with \(x^2-6x+5=0\) GMAT Problem Solving

Question: Which of the following equations has a root in common with\(x^2-6x+5=0\)?

A. \(x^2+1=0\)
B. \(x^2-x-2=0\)
C. \(2x^2-2=0\)
D. \(x^2-2x-3=0\)
E. \(x^2-10x-5=0\)

Answer:
Approach Solution (1):

First, let’s solve\(x^2-6x+5=0\)by factoring:
(x – 5) (x – 1) = 0
Therefore, x = 5 or x = 1
We can solve the equations in the given choices by factoring also. However, we see that the equation in choice A is not factorable (the equation is a sum of square), so we can start with choice B
B. (x – 2) (x + 1) = 0
X = 2 or x = -1
We see that the equation in choice B does not have a root in common with\(x^2-6x+5=0\)
\(2(x^2-1)=0\)
\(2(x-1)(x+1)=0\)
C. \(x=1orx=-1\)
We can see that the equation in choices C does have a root in common with (namely x = 1)
Correct option: C

Approach Solution (2):
It given that:
\(x^2-6x+5=0\)
\((x-1)(x-5)=0\)
Hence, roots are 1 and 5
Substitute these roots in all the given equations:
\(x^2+1=0\)
x=1;
1. \(1^2+1=1+1=2!=0\)
Not a root
x=5;
\(5^2+1=25+1=26!=0\)
Not a root
\(x^2-x-2=0\)
x=1;
2. \(1^2-1-2=2!=0\)
Not a root
x=5;
\(5^2-5-2=25-7=18!=0\)
Not a root
\(x^2-10x-5=0\)
x=1;
3. \(1^2-10*1-5=-14!=0=-2!=0\)
Not a root
x=5;
\(5^2-10*5-5=25-55=-30!=0\)
Not a root
\(2x^2-2=0\)
x=1;
4. \(2*1^2-2=2-2=0=0\)
1 is a root
We can stop here
x=5;
\(2*5^2-2=50-2=48!=0\)
Not a root
\(x^2-2x-3=0\)
x=1;
5. \(1^2-2*1-3=-3!=0\)
Not a root
x=5;
\(5^2-2*5-3=12!=0\)
Not a root
Correct option: C

Approach Solution (3):
\(x^2-6x+5=0\)
x = 5 or 1
3.\(2x^2-2=0\)
x = 1 or -1
Root x = 1 common
Correct option: C

“Which of the following equations has a root in common with\(x^2-6x+5=0\)?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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