What are the Values of P So that the Equation P x(x – 3) + 9 = 0 GMAT Problem Solving

Question: What are the values of P so that the equation P x(x – 3) + 9 = 0 has equal roots?

1. 0 and -4
2. 0 and -2
3. 0 and -1
4. 0 and 1
5. 0 and 4

Answer:
Solution with Explanation:
Approach Solution (1):

For equal roots, \(b^2=4ac\) if equation is \(ax^2+bx+c=0\)

\(Px(x–3)+9=0\)
i.e., \(Px^2−3Px+9=0\)
i.e., \((−3P)^2=4∗P∗9\)
 \(9P^2=4∗P∗9\)
 \(P^2=4∗P\)
 P = 0
or
 P=4

Correct Option: E

Approach Solution (2):

Simplifying the equation, we have:

\(Px^2 - 3Px + 9 = 0 → Eq. (1)\)

If the equation has the equal roots, then it can be factored as:

\((x\sqrt P - 3)^2 = 0\)

Expanding this equation, we have:

\(Px^2 - 6x\sqrt P + 9 = 0 → Eq. (2)\)

From Eq.(1) and Eq.(2), we see that:

\(3P = 6\sqrt P\)

\(P = 2\sqrt P\)

\(P^2 = 4P\)

\(P^2 - 4P = 0\)

P(P - 4) = 0
P = 0
or
P = 4

Correct Option: E

“What are the values of P so that the equation P x(x – 3) + 9 = 0 has equal roots?”- 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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