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In x-y plane, does parabola y=ax^2+bx+c intersect to x-axis?  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 59% (01:04) correct 41% (01:23) wrong based on 87 sessions

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In x-y plane, does parabola y=$$ax^2$$+bx+c intersect to x-axis?

1) b= -2
2) c<0

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Re: In x-y plane, does parabola y=ax^2+bx+c intersect to x-axis?  [#permalink]

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itisSheldon wrote:
In x-y plane, does parabola y=$$ax^2$$+bx+c intersect to x-axis?

1) b= -2
2) c<0

In case of a parabola(quadratic expression) ax^2 + bx + c:
If parabola intersects x-axis at two distinct points, it means the quadratic expression has two distinct real roots, and this happens when (b^2 - 4ac) > 0
If parabola intersects x-axis at one point only, it means the quadratic expression has one real root, and this happens when (b^2 - 4ac) = 0
If parabola does not intersect x-axis at all, it means the quadratic expression has no real root, and this happens when (b^2 - 4ac) < 0

So we need to know the sign of (b^2 - 4ac).

Statement 1
b is negative, but we don't know anything about a and c. Not sufficient.

Statement 2
c is negative, but we don't know anything about a and b. Not sufficient.

Combining the two statements,
b and c are negative, but we don't know the sign of a.
If a is positive, then - 4ac will be positive, and then (b^2 - 4ac) will also be positive and the parabola will intersect x-axis.
If a is negative, then -4ac will be negative, and (b^2 - 4ac) could be either negative or positive or 0 depending on the magnitudes of a, b, c. So we cant say whether parabola will intersect x-axis or not.

So not sufficient to determine.

e-GMAT Representative V
Joined: 04 Jan 2015
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In x-y plane, does parabola y=ax^2+bx+c intersect to x-axis?  [#permalink]

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SOLUTION

In $$x-y$$ plane, the parabola intersects the $$x$$-axis if the equation of the parabola has real roots.
Thus, we have to find:
• Whether $$y=ax^2+bx+c$$ has real roots or not.

Per our conceptual understanding of quadratic equation, a quadratic equation has real roots if the value of discriminant is greater or equal to zero.
For equation $$y= ax^2+bx+c$$, the value of $$D$$ is:
• $$D= b^2-4ac$$
Thus, we have to find if the value of $$(b^2-4ac) >=0$$.
Statement-1 "$$b= -2$$"

Since we don’t know the value of $$a$$ and $$c$$, statement 1 alone is not sufficient to answer the question.

Statement-2 "$$c<0$$"

Let us assume $$c=-k$$.
Since we don’t know the value of $$b$$ and $$a$$, statement 2 alone is not sufficient to answer the question.

Combining both the statements:
From both the statements combined, the value of $$D$$ is:
• $$D= (-2)^2 – 4*a*(-k)$$
• $$D= 4+4ak$$
We still do not have the value of “$$a$$”.
Hence, “Statement (1) and (2) together are not sufficient to answer the question”.

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Re: In x-y plane, does parabola y=ax^2+bx+c intersect to x-axis?  [#permalink]

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itisSheldon wrote:
In x-y plane, does parabola y=$$ax^2$$+bx+c intersect to x-axis?

1) b= -2
2) c<0

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

The question asks if the discriminant b^2 - 4ac ≥ 0.

Since we have 3 variables (a, b and c) and 0 equations, E is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)

If $$a = 1, b = -2, c = -1$$, then $$b^2-4ac = (-2)^2-4 \cdot 1 \cdot (-1) = 4 + 4 > 0$$ and it has an intersection.
If $$a = -2, b = -2, c = -1$$, then $$b^2-4ac = (-2)^2-4 \cdot (-2) \cdot (-1) = 4 - 8 < 0$$ and it doesn't an intersection.

Since both conditions together do not yield a unique solution, they are not sufficient.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
_________________ Re: In x-y plane, does parabola y=ax^2+bx+c intersect to x-axis?   [#permalink] 30 Nov 2019, 00:09
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