SOLUTION: for what value of a does the quadratic function f(x)=ax^2-6x+3 have no x-intercepts?

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Question 1089048: for what value of a does the quadratic function f(x)=ax^2-6x+3 have no
x-intercepts?
Found 2 solutions by ikleyn, MathLover1:
Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.
The necessary and sufficient condition for it is that the discriminant of the equation must be negative:

d = b%5E2+-4ac < 0, i.e.

(-6)^2 - 4*a*3 < 0,

36 < 12a,

a > 3.

See the plots below for a = 2, 3 and 4.




Plot y1 = 2x%5E2+-+6x+%2B3 (red), y2 = 3x%5E2+-+6x+%2B3 (green), y3 = 4x%5E2+-+6x+%2B3 (blue)



Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
An x-intercept is the point where a parabola crosses the x-axis. This point is also known as a zero, root, or solution.
the expression b%5E2+%96+4ac, called the "discriminant", and we can use it to find out what kind and how many roots the quadratic function have
rule:
if b%5E2+-4ac+%3E+0, Discriminant is greater than zero, Positive Discriminant:Two+Real Solutions
if b%5E2-4ac+=+0, Discriminant is equal to zero. One+Real Solution
if b%5E2+-4ac+%3C+0, Discriminant is less than zero ,Negative Discriminant: No+Real Solutions => means no x-intercepts

so, you will use b%5E2+-4ac+%3C+0 for f%28x%29=ax%5E2-6x%2B3
since a=a, b=-6, and c=3, you have
%28-6%29%5E2+-4a%2A3%3C+0
36+-12a%3C+0
36+%3C+12a
36%2F12+%3C+a
a%3E3
check:
if a%3E3, we can use a=4 and see the graph of f%28x%29=4x%5E2-6x%2B3
+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+4x%5E2-6x%2B3%29+
as you can see, there is no x-intercept