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Question 115371: The x-axis is tangent to the graph of the function y=x^2+ax+1 if and only if a=? or ?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! I'm going to presume that you are familiar with the quadratic formula. It says that if you
have a quadratic equation of the standard form:
.
ax^2 + bx + c = 0
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that the values of x that make this equation true are given by:
,
.
The term under the radical sign is called the discriminant. If it is a positive
value, the
quadratic formula will give you two real and unequal values for x. This means that the graph
of the quadratic function you are given crosses the x-axis at two points. If the discriminant
is a negative number, you have to take the square root of a negative number to solve the
values of x that satisfy the equation. This means that you end up with complex values for
x and indicates that the graph does not cross or touch the x-axis at all. Finally, if the
discriminant is zero, it means that there is only one value for x that satisfies the quadratic
equation, and that means the graph of the quadratic function just touches the x-axis at
one point, the point of tangency. (That point is the vertex ... the peak or the lowest
point ... of the parabolic graph).
.
Now I'm going to restate your problem just a little to prevent confusion. I'm going to make
it read:
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"The x-axis is tangent to the graph of the function y=x^2+bx+1 if and only if b=? or ?"
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This makes your function match the notation of the standard form in that the multiplier
of x is now called b in both the standard form and in your problem.
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If you set y equal to zero in your function, your problem then becomes:
.
.
Compare this to the standard form and you see that "a" (the multiplier of the x^2 term) is 1 ...
and "b" (the multiplier of the x term) is still b in your problem ... and "c" (the constant)
is +1 in your problem.
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Now if you were to apply the quadratic formula to your problem you would say that the
values of x that satisfy the formula are given by:
.
.
If you substitute 1 for "a", "b" for "b", and 1 for "c" you get that the values of x that
satisfy your problem are:
.
.
Notice that the term in the radical is:
.
.
This is the discriminant and if it equals zero, the graph is tangent to the x-axis.
So set this discriminant for your problem equal to zero and you have the equation:
.
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Multiply out the terms in parentheses and you get:
.
.
Get rid of the -4 on the left side by adding 4 to both sides and you have:
.
.
Solve for b by taking the square root of both sides and you have two possible answers:
.
 and 
.
This means that the graph of the function you were given originally ... that is the graph
of the function:
.
.
will be tangent to the x-axis at one point if b equals +2 or if b equals -2.
.
So the graphs of:
.
.
and
.
.
are each tangent to the x-axis at only 1 point.
.
Hope this helps you to understand the problem a little better and shows you the value of
using the discriminant in analyzing the position of a graph of a quadratic function relative
to the x-axis.
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