Quadratics: Quadratic Equations

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Discussion Overview

The discussion revolves around the quadratic equation x^2 + px + 2p = 0, specifically focusing on finding the discriminant and determining the values of p that yield different types of solutions (two solutions, no solutions, and one solution). The scope includes mathematical reasoning and problem-solving related to quadratic equations.

Discussion Character

  • Mathematical reasoning
  • Homework-related
  • Exploratory

Main Points Raised

  • One participant calculates the discriminant as p^2 - 8p, but expresses uncertainty about its correctness.
  • Another participant questions the discriminant calculation, suggesting it might be p^2 - 4 - 8p, seeking confirmation and assistance with subsequent parts of the problem.
  • A different participant provides a general derivation of the quadratic formula, emphasizing the importance of the discriminant's value in determining the nature of the solutions.
  • There is a request for participants to show their working out to facilitate better assistance and understanding.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the correct form of the discriminant or the subsequent steps to solve for the values of p. Multiple interpretations and approaches are presented, indicating ongoing uncertainty and exploration.

Contextual Notes

There are unresolved aspects regarding the calculations of the discriminant and the conditions for the number of solutions, with participants expressing varying degrees of understanding and confidence in their approaches.

James400
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Consider the quadratic equation x^2+px+2p=0

a. Find the discriminant.

b. Find the values of p for which there are 2 solutions.

c. Find the values of p for which there are no solutions.

d. Find the value of p for which there is 1 solution.

Please show working out! Thanks.
 
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James400 said:
Please show working out! Thanks.

How about your working out! :). How can we help if we don't know what you are struggling with :p.
 
For a. Find the discriminant:
=b^2-4ac
=p^2-8p

That's is as far as I have gotten, although I think that is right for a. correct me if I am wrong. So I just need to solve the others, although I have no idea how to do so...
 
James400 said:
Consider the quadratic equation x^2+px+2p=0

a. Find the discriminant.

b. Find the values of p for which there are 2 solutions.

c. Find the values of p for which there are no solutions.

d. Find the value of p for which there is 1 solution.

Please show working out! Thanks.

Is anyone able to confirm whether a. is p^2-4-8p and help solve b through to d?
 
Ok, let's start from scratch.

$$ax^2+bx+c=0$$

$$a(x^2+\frac bax)=-c$$

$$a\left(x+\frac{b}{2a}\right)^2-\frac{b^2}{4a}=-c$$

$$a\left(x+\frac{b}{2a}\right)^2=\frac{b^2}{4a}-c$$

$$\left(x+\frac{b}{2a}\right)^2=\frac{b^2}{4a^2}-\frac ca$$

$$x+\frac{b}{2a}=\pm\sqrt{\frac{b^2}{4a^2}-\frac{4ac}{4a^2}}$$

$$x=-\frac{b}{2a}\pm\sqrt{\frac{b^2}{4a^2}-\frac{4ac}{4a^2}}$$

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Now plug your values for $a$, $b$ and $c$ into $b^2-4ac$.

Where is $b^2-4ac>0$? Less than $0$? Equal to $0$? Do you understand the relevance of these relationships?
 

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