If (1−p) is a Root of Quadratic Equation x^2+px+(1−p)= 0 Then GMAT Problem Solving

Question: If (1−p) is a root of quadratic equation \(x^2\)+px+(1−p)=0 then its roots are

0, -1
-1, 1
0, 1
-1, 2
2, 3

Correct Answer: A

Solution and Explanation
Approach Solution 1:
Given:

(1−p) is a root of quadratic equation \(x^2\)+px+(1−p)=0

Find out:

The roots of \(x^2\)+px+(1−p)=0

If (1 − p) is a root, then x = (1 − p) is a solution to the equation x² + px + (1 − p) = 0
Now, if we replace x with (1 - p),
we get: (1 - p)² + p(1 - p) + (1 - p) = 0
After we factor out the (1 - p) to get: (1 - p)[(1 - p) + p + 1] = 0
If we simplify: (1 - p)[2] = 0
So, p = 1

If p = 1, our equation, x² + px + (1 − p) = 0, becomes x² + (1)x + (1 − 1) = 0
Simplify: x² + x = 0
Factor: x(x + 1) = 0
So, EITHER x = 0 OR x = -1

Hence, the correct answer is A

Approach Solution 2:
Given:

(1−p) is a root of quadratic equation x2+px+(1−p)=0

Find out:

The roots of x2+px+(1−p)=0

If \(x^2\)+px+(1−p)=0

=>\(x_1\)= (1-p)

In that case,

=> \((1-p)^2\)+p(1–p)+(1–p)=0
=> (1—p)(1–p+p+1)=0
=> 2(1–p)=0
=> p=1

Now considering:
=>\(x^2\)+x+0=0
=>x(x+1)=0
=>x=-1, x=0
Hence, the correct answer is A

Approach Solution 3:
(1-p) is a root
Therefore, (1-p)^2+p(1-p)+1-p= 0
=> (1-p) (1-p+p+1)= 0
=> (1-p) (2)= 0
=> p= 1
x^2+x= 0

One root= 0 and another root = -1
Therefore, roots are 0 and -1

"If (1−p) is a root of quadratic equation \(x^2\)+px+(1−p)=0 then”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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