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Graph of quadratic equation

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by vipulgoyal » Mon Feb 03, 2014 2:49 am
we know that sum of the roots are -b/a and product of the roots are c/a

as given putting y=o , n amd m are two roots with n is bigger root

(n-m)^2 = (n+m)^2 - 4nm
y = x^2 + ax + b with given equation sum of root is -a and product of root is b

(n-m)^2 = a^2-4b
from 1 : a^2-4b = 4
hence n-m = 2
from 2: we only know ptoduct; we dont know product hence insuff
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by GMATGuruNY » Mon Feb 03, 2014 4:29 am
gmattesttaker2 wrote: If the graph of y = x^2 + ax + b passes through the points (m, 0) and (n, 0), where m < n, what is the value of n - m ?

(1) 4b = a^2 - 4

(2) b = 0
What is the value of n-m?
(m,0) and (n,0) are the two points at which y=0 and the graph intersects the x-axis.
Thus, m and n are the x-intercepts of the graph.
Since m<n, n-m > 0.
Question rephrased: What is the positive difference between the x-intercepts?

Statement 1: 4b = a²-4
Test easy cases.

Case 1: a=0
If a=0, we get:
4b = 0²-4
b=-1.

Substituting a=0 and b=-1 into y = x² + ax + b, we get:
y = x² + 0x - 1.
y = x² - 1.
Here, y=0 when x=-1 or x=1.
Thus, the x-intercepts are -1 and 1.
Result:
Positive difference between the x-intercepts = 1 - (-1) = 2.

Case 2: a=2
If a=2, we get:
4b = 2²-4
b=0.

Substituting a=2 and b=0 into y = x² + ax + b, we get:
y = x² + 2x + 0
y = x(x+2).
Here, y=0 when x=0 or x=-2.
Thus, the x-intercepts are -2 and 0.
Result:
Positive difference between the x-intercepts = 0 - (-2) = 2.

The cases above illustrate that -- given the constraint in statement 1 -- the positive difference between the x-intercepts must be 2.
SUFFICIENT.

Statement 2: b=0
In Case 2, b=0 and a=2.
In this case, the positive difference between the x-intercepts is 2.

Case 3: b=0 and a=1
Substituting a=1 and b=0 into y = x² + ax + b, we get:
y = x² + 1x + 0
y = x(x+1).
Here, y=0 when x=0 or x=-1.
Thus, the x-intercepts are -1 and 0.
Result:
Positive difference between the x-intercepts = 0 - (-1) = 1.

Since the positive difference is not the same value in each case, INSUFFICIENT.

The correct answer is A.
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