determine the number of real solutions by examining the graph of y= ax^2+bx+c=0?

Question

The question is asking me to "determine the number of real solutions by examining the graph of y= ax^2+bx+c=0"

I have two problems

1) 1.44x^2-8.4x+12.25=0
2) -x^2+5x-6.1=0

How do I tell the number of solutions based on the graph? Can anyone help me solve these problems or give me a link to see how to do this? Thank you very much.

Answer 1

Edit: Added the plots of your problems in the applicable sections below:



The Discriminant:

The expression in the square root, D = b^2 - 4ac , is called the discriminant, because its value can be used to discriminate between the different types of solutions that are possible:

1) D > 0 results in two real solutions:

a) If D is a perfect square (like 4, 9 64, 100, etc.) then you get two rational solutions.

b) If D is a not perfect square then you get two irrational solutions. Example: 2)

http://www.wolframalpha.com/input/?i=plot++-x^2%2B5x-6.1%3D0



2) If D = 0 then you get a single real solution. Example: 1)

http://www.wolframalpha.com/input/?i=plot+1.44x^2-8.4x%2B12.25%3D0



3) If D < 0 then you get two complex solutions. (Note: you only get imaginary solutions if the discriminant is negative and b = 0!)

Answer 2

For any problem like this, you go by the number of places where the graph touches or crosses the x-axis. Three possibilities:

--- crosses in 2 places (2 real solutions)

--- touches at the vertex (1 real solution)

--- does not touch or cross (0 real solutions)

Answer 3

If you have the graph, just see how many times the line crosses or touches the x axis.

Answer 4

any polynomial of degree 2 can give you at most 2 solutions.