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Jun 21, 2022 6:45 pm
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
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Question Stats:
66% (01:39) correct
34%
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wrong
based on 349
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In the equation \(x^2+kx+1=0\), x is a variable and k is a constant. If the quadratic equation has two distinct real roots, which of the following could be true?
I. k = 0
II. k = −1
III. k = −3
A. I only
B. III only
C. I and II only
D. II and III only
E. I and III only
I. k = 0
II. k = −1
III. k = −3
A. I only
B. III only
C. I and II only
D. II and III only
E. I and III only
Jun 22, 2022 12:28 am
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Generally, the concept of Delta Δ is not approached in the GMAT studies, but for this question, in my opinion is the fastest way to “kill” this question.
Quadratic equation is given by ax^2 +bx + c = 0 and Δ = b^2 - 4ac.
Two distinct real roots demands Δ > 0.
Quadratic equation is given by ax^2 +bx + c = 0 and Δ = b^2 - 4ac.
Two distinct real roots demands Δ > 0.
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Sep 15, 2026 1:54 am
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Bunuel
One way to solve a question like this without the quadratic formula is by "completing the square".
Completing the square involves using one of the following algebraic identities to solve:
\((a - b)^2 = a^2 - 2ab + b^2\) or \((a + b)^2 = a^2 + 2ab + b^2\)
Statement I. k = 0
\(x^2 + 1 = 0\)
\(x^2 = -1\)
No real number squared will be negative. Statement I is out.
Statement II. k = -1
\(x^2 - x + 1 = 0\)
\(x^2 - x = -1\)
Note that if we divide k by 2 and square it, we get:
\((-1/2)^2 = 1/4\)
Now add 1/4 to both sides:
\(x^2 - x + 1/4 = -3/4\)
\((x - 1/2)^2 = -3/4\)
Again, no real number squared will be negative. Statement II is out. The answer must be B. Let's confirm by trying Statement III:
Statement III. k = -3
\(x^2 - 3x + 1 = 0\)
\(x^2 - 3x = -1\)
Again, divide k by 2 and square it:
\((-3/2)^2 = 9/4\)
Add 9/4 to both sides:
\(x^2 - 3x + 9/4 = -1 + 9/4\)
\((x - 3/2)^2 = 5/4\)
Because the right side is positive, this equation is solvable with two real solutions.
Unless you're already comfortable with the quadratic formula, knowing how to complete the square may be preferable to needing a formula that is easy to misremember. It's unlikely you'll need either on the GMAT.
