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

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Manager
Joined: 03 Mar 2018
Posts: 170
In x-y plane, does parabola y=ax^2+bx+c intersect to x-axis? [#permalink]

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06 Mar 2018, 11:40
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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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Joined: 22 Aug 2013
Posts: 1209
Location: India
Re: In x-y plane, does parabola y=ax^2+bx+c intersect to x-axis? [#permalink]

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07 Mar 2018, 02:47
1
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.

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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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19 Mar 2018, 13:06

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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In x-y plane, does parabola y=ax^2+bx+c intersect to x-axis?   [#permalink] 19 Mar 2018, 13:06
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