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21 Apr 2006, 03:16
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Does the line represented by y=ax^2+bx+c intersect with X-axis?
1)a>0
2)c<0
1)a>0
2)c<0
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21 Apr 2006, 11:18
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The line would intersect with x asis when y becomes 0, ie, when
ax^2 + bx + c = 0 holds true.
From 1, a > 0.
Thus x^2 is not a zero entity. Thus, the line would have a exponential component in the graph.
Also, x^2 would always be positive, and so ax^2 is a positive entity.
But we don't know the signs of b and c and therefore don't know if this expression turns 0.
From 2, c < 0. Thus, for x = 0, the expression is -ve. If the expression is positive for some values of x, then the expression would also be 0 before turning -ve. but we can't assume that, since we don't know the value of a and b.
Combining, we don't know the value of b. But for big enough values of x, ax^2 would be bigger than bx + c and the expression would be +ve.
Therefore, C.
Can someone solve it in a less cumbersome way?
ax^2 + bx + c = 0 holds true.
From 1, a > 0.
Thus x^2 is not a zero entity. Thus, the line would have a exponential component in the graph.
Also, x^2 would always be positive, and so ax^2 is a positive entity.
But we don't know the signs of b and c and therefore don't know if this expression turns 0.
From 2, c < 0. Thus, for x = 0, the expression is -ve. If the expression is positive for some values of x, then the expression would also be 0 before turning -ve. but we can't assume that, since we don't know the value of a and b.
Combining, we don't know the value of b. But for big enough values of x, ax^2 would be bigger than bx + c and the expression would be +ve.
Therefore, C.
Can someone solve it in a less cumbersome way?
21 Apr 2006, 11:29
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kapslock
Just remember the properties:
If a>0 & c>0 parabola opens upwards and the vertex of the parabola is above x axis and the curve will not intersect the x-axis
If a>0 & c <= 0, the parabola opens upwards and the vertex of parabola is touching x-axis or is below the x-axis. Hence it has one or two points.
The opposite is true for a<0... parabola opens downwards....
You can verify this by putting x=0, If c is -ve, kowning that a>0, the parabola should intersect the x-axis at 2 points!
