Values in Quadratic Equation with 2 different roots

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Homework Help Overview

The discussion revolves around determining the intervals of possible values for p in the quadratic equation (p-1)x² + 4x + (p-4) = 0, specifically focusing on when the equation has two different roots.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the conditions under which a quadratic equation has two different roots, questioning the role of the discriminant and how to apply the quadratic formula. There are attempts to substitute specific values for p to observe the behavior of the roots.

Discussion Status

The conversation includes various attempts to understand the problem, with some participants suggesting the use of the quadratic formula and others questioning the assumptions about the roots. There is a recognition of the importance of the discriminant in determining the nature of the roots, and some guidance has been offered regarding how to set up the inequality for p.

Contextual Notes

Participants are navigating the problem with varying levels of familiarity with quadratic equations and the discriminant, leading to discussions about the necessity of graphing versus analytical methods. There is an emphasis on understanding the conditions for different types of roots without reaching a final consensus on the exact intervals for p.

FlopperJr
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Homework Statement



Find the intervals of all possible value of p which the equation equation: (p-1)x^2+4x+(p-4)=0 has two different roots.



Homework Equations



ax^2+bx+c>0 ??

The Attempt at a Solution



(p-1)x^2+4x+(p-4)>0 ??

How would I go about solving this?
Is two roots >0 or =0??

Could I have an example so that I could then solve the problem on my own or steps I need to take in order to solve for the values of p.
 
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Can you solve a quadratic equation? What formula do you use for the solution? In case of p=2 you have the quadratic equation x^2+4x-2. What are the roots?

ehild
 
yes but I don't know with (p-1) and (p+4)

-b+/- Squareroot b^2 - 4ac over 2a ?

Do i plug it in that?
 
Yes. Use the quadratic formula with a = p - 1, b = 4, and c = p - 4.
 
ohh okay. I used a pgraphing calculator and think I got the answer.
p< or equal to 4

and
p>1
 
Are you sure? What are the roots if p=0.5? When p=4.5?

ehild
 
Ohh. I see. I believe my answer now is p<5 and p>0.
 
It is better. But why do you think so?
When has a quadratic equation two different roots?

ehild
 
When it intersects the x axis. It intersects at two different points. So I just plugged in values for p until I began to see two different roots and then narrowed down my answer.
 
  • #10
FlopperJr said:
When it intersects the x axis. It intersects at two different points. So I just plugged in values for p until I began to see two different roots and then narrowed down my answer.

Yu can not get the exact solution from a graph. By the way, people could (and can) solve quadratic equations without graphing calculators.
Use the formula that you copied already for the solution of the quadratic equation

ax2+bx+c=0

[tex]x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

Because of the ± in front of the square root, the quadratic equation has two real roots if the expression under the square root (the discriminant) is greater than zero. There is a single solution if it is equal to zero, and no real solution if the discriminant is negative.
So you need to find those p values which make the discriminant greater than zero:
[tex]b^2-4ac>0[/tex]
Plug in the expressions for a, b, and c, and solve the inequality for p.

ehild
 
  • #11
Thank you so much for your help! i could not have solved it without your help. I get it. Now let's just see how do on my test this Wednesday.
 
  • #12
Good luck! ehild
 

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