Question: Does the graphical representation of the quadratic function f(x) = y = ax^2 + c intersect with the x - axis?

(1) a < 0
(2) c > 0

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

Solutions and Explanation

Approach Solution : 1

Essentially, the question asks whether y is possible for some value(s) of x. Consequently, whether ax^2+c=0 has true roots.

a(x^2) + c = 0
=>
x^2 = -(c/a). I

If -(c/a) is greater than or equal to zero, then the above equation will have real roots.

Statement - 1 : a < 0

Information about c is not given.
As a result, this statement is not sufficient.

Statement - 2 : c > 0

Information about a is not given.

As a result, this statement is not sufficient.

By combining both the statements, we get a<0 and c>0.

It means that -(c/a) = −(positive/negative) = −negative = positive > 0

Therefore this is sufficient.

Correct Answer: (C)

Approach Solution : 2

Avoiding imaginary roots and simplifying the equation y = a(x^2) + c to get x^2= -(c/a) for the X-axis intercept is the crucial component to solving the problem.

Statement - 1 : a < 0

Only when c > 0 can we avoid imaginary roots. If c<0, it becomes imaginary.

This statement is insufficient because both scenarios are plausible.

Statement - 2 : c > 0

Only when a<0 can we avoid imaginary roots. If a>0, it becomes imaginary

It is not possible to find out x-intercept

Therefore this statement is insufficient.

By combining both the statements, we get the value c>0 and a<0 which is required.

Therefore this is sufficient.

Correct Answer: (C)

Approach Solution : 3

Let us use the graph approach

Statement - 1 : a < 0

Parabola* is described as having a downward slope.

But since the vertex is unknown, it may or may not cross the x-axis.

As a result, this statement is not sufficient.

Statement - 2 : c > 0

Since the vertex could be above x=0 but the y intercept is positive, there won't be any solutions for x.

As a result, this statement is not sufficient.

Let us combine both the statements.

We can infer that since the curve slopes downward and the 'y' intercept is positive, there must be at least one solution for 'x'.

Therefore this is sufficient.

Correct Answer: (C)

“Does the graphical representation of the quadratic function” - is a topic of the GMAT Quantitative reasoning section of GMAT. GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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