Factoring quadratic equation (with trig identities used)

Click For Summary

Discussion Overview

The discussion revolves around the possibility of factoring a quadratic equation of the form \( a \sin^2 x - b \sin 2x + c \). Participants explore the challenges posed by the term \( \sin 2x \) and consider various approaches to manipulate or solve the equation using trigonometric identities.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the feasibility of factoring the expression due to the presence of \( \sin 2x \), which expands to \( 2 \sin x \cos x \).
  • Another participant suggests that the equation can be rewritten as \( a \sin^2(x) + c = 2b \sin(x) \sqrt{1 - \sin^2(x)} \) after rearranging, leading to a quadratic form in \( \sin^2(x) \).
  • A different participant proposes using trigonometric identities to express \( \sin^2 x \) in terms of \( \cos 2x \), suggesting this could lead to a solvable equation of the form \( p \cos 2x - b \sin 2x + q = 0 \).
  • There is a repeated inquiry into the context of the problem, with one participant noting that the original expression lacks an equality sign, implying a potential misunderstanding.

Areas of Agreement / Disagreement

Participants express differing views on the possibility of factoring the quadratic equation, with some proposing methods to manipulate the equation while others remain skeptical about the initial formulation and its context. No consensus is reached regarding the best approach or the validity of the original expression.

Contextual Notes

Participants note the potential confusion stemming from the lack of an equality sign in the original expression, which may affect the interpretation of the problem. Additionally, the discussion highlights the dependence on trigonometric identities and the assumptions made in manipulating the equation.

stratz
Messages
23
Reaction score
0
Is it possible to factor a quadratic equation along the lines of asin^2x -bsin2x+c ? If so, how? The sin2x seems to be a problem since when expanded it becomes 2sinxcosx, but I'm wondering if it is possible, and how it would be done?

Thanks in advance.
 
Mathematics news on Phys.org
stratz said:
Is it possible to factor a quadratic equation along the lines of asin^2x -bsin2x+c ? If so, how? The sin2x seems to be a problem since when expanded it becomes 2sinxcosx, but I'm wondering if it is possible, and how it would be done?
I don't see any way of doing it. What's the context of this problem? Could there be a mistake leading up to what you show?

BTW, what you show isn't an equation -- there's no =.
 
a sin2(x) - b sin(2x) + c = 0 (I guess that is what you meant) can be written as ##a \sin^2(x) + c = 2 b \sin(x) \sqrt{1-\sin^2(x)}##, after squaring both sides you get a quadratic equation in sin2(x).
 
stratz said:
Is it possible to factor a quadratic equation along the lines of asin^2x -bsin2x+c ? If so, how? The sin2x seems to be a problem since when expanded it becomes 2sinxcosx, but I'm wondering if it is possible, and how it would be done?

Thanks in advance.

Is the object to solve a \sin^2x - b \sin 2x + c = 0 for x? If so, use the identities <br /> \cos^2 x + \sin ^2 x = 1 \\<br /> \cos^2 x - \sin^2 x = \cos 2x<br /> to express \sin^2 x in terms of \cos 2x; then you'll have something of the form p \cos 2x - b \sin 2x + q = 0 which I hope you know how to solve.
 

Similar threads

MHB Solve Quadratic Equation: Find c-a Given p and q
  • · Replies 4 ·
Replies
4
Views
2K
High School Can negative roots of a quadratic equation for √E be physically acceptable?
  • · Replies 2 ·
Replies
2
Views
2K
High School I badly with general answers to trig equations. (Using identities)
  • · Replies 32 ·
2
Replies
32
Views
3K
MHB Confused about a couple of trig. identities.
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Proof of PSF quadratic factorising
  • · Replies 1 ·
Replies
1
Views
4K
High School How Can Trigonometry Calculate Distances in Surveying?
  • · Replies 5 ·
Replies
5
Views
3K
Comp Sci Octave coding -- solving a quadratic equation and then using the roots
  • · Replies 5 ·
Replies
5
Views
3K
Graduate A Generalized Quadratic Formula
  • · Replies 3 ·
Replies
3
Views
5K
High School English Translation of Italian Lecture Notes: Equations & Solutions
  • · Replies 11 ·
Replies
11
Views
2K
High School Cartesian Space vs. Euclidean Space
  • · Replies 10 ·
Replies
10
Views
3K