'least value of a if x^2-4x+6-a is never a negative'....How to solve??? Please help!

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minisue1

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If anyone can help me out on this please.... I have no idea how to go about this!
 
minisue1 said:
If anyone can help me out on this please.... I have no idea how to go about this!
Click to expand...
Let's start by getting some preliminaries out of the way. I suspect you can easily answer the questions below.

An equation with the form \(\displaystyle y = px^2 + qx + r\) is what kind of equation and has how many distinct roots?

The graph of an equation with the form \(\displaystyle y = px^2 + qx + r\) is what kind of curve and has how many x-intercepts?

This one may be slightly harder to answer.

If the graph of an equation with the form \(\displaystyle y = px^2 + qx + r\) is always positive, how many x-intercepts does it have?
 
Complete the square! \(\displaystyle (x- b)^2= x^2- 2bx+ b^2\). Comparing that to \(\displaystyle x^2- 4x+ 6- a\), you should see that "2bx" is "4x" so 2b= 4 and b= 2. Then\(\displaystyle b^2= 2^2= 4\) so that \(\displaystyle x^2- 4x+ 4\) is a 'perfect square'. 6= 4+ 2 so we can write \(\displaystyle x^2- 4x+ 6- a= (x^2- 4x+ 4)+ (2- a)= (x- 2)^2+ (2- a)\). Now a square is never negative but \(\displaystyle (x- 2)^2\) can be 0 at x= 2. That is, 2- a is the lowest possible value. What must a be equal to so that 2- a is never negative?
 
JeffM said:
Let's start by getting some preliminaries out of the way. I suspect you can easily answer the questions below.

An equation with the form \(\displaystyle y = px^2 + qx + r\) is what kind of equation and has how many distinct roots?

The graph of an equation with the form \(\displaystyle y = px^2 + qx + r\) is what kind of curve and has how many x-intercepts?

This one may be slightly harder to answer.

If the graph of an equation with the form \(\displaystyle y = px^2 + qx + r\) is always positive, how many x-intercepts does it have?
Click to expand...

Hello, Minisue1,

By any chance, has your class been discussing "discriminants" (b^2 - 4ac) lately? View the information here:

http://www.mathwarehouse.com/quadratic/discriminant-in-quadratic-equation.php

to help you answer Jeff's questions. This should help you see the "big picture" and understand the approach to the problem.
 
If your class has discussed shifting parabolas up or down, then one graphical approach is to realize that the greatest value of a is the perpendicular distance from the x-axis to the vertex point. I don't know what you already know. :cool:
 
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