Describing the Graph of an Equation

Date: 1/27/96 at 20:35:4
From: Gorav Khanna
Subject: graph of an equation

Could I please have some help with this question? Thanks...

Which of the following is true of the graph of the equation 
y=2x^2 - 5x + 3 ? 

   A. It is tangent to the x-axis
   B. It intersects the x-axis at only two distinct points
   C. It intersects the x-axis at more than two distinct points
   D. It lies completely below the x-axis
   E. It lies completely above the x-axis

Date: 4/28/96 at 19:37:37
From: Doctor Steven
Subject: Re: graph of an equation

We can analyze this to see what we can throw out right away.

The parabola opens upward since the coefficient of x^2 is positive 
so therefore D cannot be true.

It is only a second degree equation, so it could not intersect the 
x-axis more than twice. So C cannot be true.

So now we have three choices: A, B, or E.

If it's A, then when y = 0 there can only be one value of x that will 
satisfy the equation.

If it's B, then when y = 0, there will be 2 values of x that satisfy 
the equation.

If it's E, then when y = 0, there are no values of x that satisfy 
the equation.

To find out which it is we will set y to zero and look for solutions.

So 2x^2 - 5x + 3 = 0.

By the quadratic equation we get:

   x = [5 +- Sqrt(25 - 24)]/4 = [5 +- 1]/4 = 6/4 or 4/4 = 3/2 or 1.

There are two values of x which satisfy the equation, so B is the 
correct answer.

Hope this helps.

-Doctor Steven,  The Math Forum