Solve Quadratic Function: Find x-Intercepts & Just Touches x-Axis

In summary, the conversation discusses finding the values for b in a quadratic fraction in order to determine the behavior of its graph. The values for b are determined by the discriminant b^2-4ac, where a and c are known coefficients. When the discriminant is positive, the graph will have two x-intercepts. When it is zero, the graph will just touch the x-axis. And when it is negative, the graph will not touch or intersect the x-axis. The values for b that lead to these conditions are b > √40 for two x-intercepts, b = √40 for one x-intercept, and b < √40 for no x-intercepts.
  • #1
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1. A quadratic fraction is of the form f(x)= 2x^2+bx+5
Find the values for b where the graph of f(x):
1.just touches the x-axis
2. has two x-intercepts
3.does not cut or touch the x-axis


I've tried using the quadratic formula but i have no idea how to do it when there are two unknowns. For number one i thought it must be when y=0 and I've tried substituting other numbers in but other than that i can't start. Can you please start me off?
 
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  • #2
When you apply the quadatic formula what are you getting? If [tex] x_{12} = \frac{-b \pm \sqrt{b^2-4ac}}{2a} [/tex] then what does it mean if [tex] x_{12} [/tex] is complex? What about if [tex] x_{12} [/tex] is real? What condition yields a [tex] x_{12} [/tex] as complex? and which condition yields [tex] x_{12} [/tex] as real?
 
  • #3
b has to be a real number but how can you work it out when there are two unknowns .. x is unknown and so is b
 
  • #4
Using the quadratic formula FrogPad posted, pay special attention to the [tex]b^2-4ac[/tex] part.
 
  • #5
You do not need to look at the complete quadratic formula, just the discriminant b2-4ac. Do you know the condition on the discriminant for the equation to have (i)two real roots; (ii) one real repeated root; (iii) no real roots (equivalent to saying the equation has complex roots)?
 
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  • #6
wellY--3 said:
b has to be a real number but how can you work it out when there are two unknowns .. x is unknown and so is b

Yes, but that is before you take into account the 3 subtasks. If a graph touches the x-axis once, how does that affect the number of real solutions to the equation?
 
  • #7
ooooooook so...
1. b=square root of 40
2 b is greater than square root of 40
3. b is less than square root of 40

is that right?
 

1. What is a quadratic function?

A quadratic function is a mathematical function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and x is the variable. This function produces a parabola when graphed and is commonly used to model real-world phenomena such as projectile motion and the shape of a satellite dish.

2. How do I find the x-intercepts of a quadratic function?

The x-intercepts of a quadratic function are the points where the graph of the function crosses the x-axis. To find these points, set the function equal to zero and solve for x. This can be done using the quadratic formula or by factoring the function. The resulting values of x are the x-intercepts.

3. What does it mean for a quadratic function to just touch the x-axis?

When a quadratic function just touches the x-axis, it means that the graph of the function intersects the x-axis at only one point. This point is known as the vertex of the parabola and is either the minimum or maximum point of the function, depending on the direction of the parabola.

4. Can a quadratic function have more than two x-intercepts?

No, a quadratic function can have a maximum of two x-intercepts. This is because the graph of a quadratic function is a parabola and a parabola only intersects the x-axis at two points at most. If a quadratic function has no real solutions, it will have no x-intercepts, and if it has one real solution, it will have one x-intercept.

5. How do I graph a quadratic function with given x-intercepts and just touches the x-axis?

To graph a quadratic function with given x-intercepts and just touches the x-axis, plot the x-intercepts on the graph and then plot the vertex at the point where the function just touches the x-axis. Use the shape of the parabola to connect these points and extend the graph as needed. You can also use the standard form of the quadratic function to determine the vertex and other key points on the graph.

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