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

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

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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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New post 07 Mar 2018, 01:47
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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.

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

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New post 19 Mar 2018, 12:06

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”.

Answer: E
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In x-y plane, does parabola y=ax^2+bx+c intersect to x-axis?   [#permalink] 19 Mar 2018, 12:06
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