Are there theorems about extraneous solutions? What causes them and how many are produced? If so, which branch of mathematics would I find such theorems (and proofs)? Same thing with missing solutions.
2026-04-03 09:24:39.1775208279
Are there theorems about extraneous solutions?
229 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in ALGEBRA-PRECALCULUS
- How to show that $k < m_1+2$?
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Finding the value of cot 142.5°
- Why is the following $\frac{3^n}{3^{n+1}}$ equal to $\frac{1}{3}$?
- Extracting the S from formula
- Using trigonometric identities to simply the following expression $\tan\frac{\pi}{5} + 2\tan\frac{2\pi}{5}+ 4\cot\frac{4\pi}{5}=\cot\frac{\pi}{5}$
- Solving an equation involving binomial coefficients
- Is division inherently the last operation when using fraction notation or is the order of operation always PEMDAS?
- How is $\frac{\left(2\left(n+1\right)\right)!}{\left(n+1\right)!}\cdot \frac{n!}{\left(2n\right)!}$ simplified like that?
- How to solve algebraic equation
Related Questions in POLYNOMIALS
- Alternate basis for a subspace of $\mathcal P_3(\mathbb R)$?
- Integral Domain and Degree of Polynomials in $R[X]$
- Can $P^3 - Q^2$ have degree 1?
- System of equations with different exponents
- Can we find integers $x$ and $y$ such that $f,g,h$ are strictely positive integers
- Dividing a polynomial
- polynomial remainder theorem proof, is it legit?
- Polyomial function over ring GF(3)
- If $P$ is a prime ideal of $R[x;\delta]$ such as $P\cap R=\{0\}$, is $P(Q[x;\delta])$ also prime?
- $x^{2}(x−1)^{2}(x^2+1)+y^2$ is irreducible over $\mathbb{C}[x,y].$
Related Questions in PROOF-EXPLANATION
- (From Awodey)$\sf C \cong D$ be equivalent categories then $\sf C$ has binary products if and only if $\sf D$ does.
- Help with Propositional Logic Proof
- Lemma 1.8.2 - Convex Bodies: The Brunn-Minkowski Theory
- Proof of Fourier transform of cos$2\pi ft$
- Total number of nodes in a full k-ary tree. Explanation
- Finding height of a $k$-ary tree
- How to get the missing brick of the proof $A \circ P_\sigma = P_\sigma \circ A$ using permutations?
- Inner Product Same for all Inputs
- Complex Derivatives in Polar Form
- Confused about how to prove a function is surjective/injective?
Related Questions in ROOTS
- How to solve the exponential equation $e^{a+bx}+e^{c+dx}=1$?
- Roots of a complex equation
- Do Irrational Conjugates always come in pairs?
- For $f \in \mathbb{Z}[x]$ , $\deg(\gcd_{\mathbb{Z}_q}(f, x^p - 1)) \geq \deg(\gcd_{\mathbb{Q}}(f, x^p - 1))$
- The Heegner Polynomials
- Roots of a polynomial : finding the sum of the squares of the product of two roots
- Looking for references about a graphical representation of the set of roots of polynomials depending on a parameter
- Approximating the first +ve root of $\tan(\lambda)= \frac{a\lambda+b}{\lambda^2-ab}$, $\lambda\in(0,\pi/2)$
- Find suitable scaling exponent for characteristic polynomial and its largest root
- Form an equation whose roots are $(a-b)^2,(b-c)^2,(c-a)^2.$
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
When learning to solve equations, the first fundamental rule is "Do the same thing to both sides". However, some times you need to be careful.
When the thing you do isn't always valid (for instance, you're dividing with something that could potentially be $0$), then you might end up losing solutions. Example: $$ x(x-5)=3x\\ x-5=3\\ x=8 $$ Dividing by $x$ lost us the solution $x=0$ (because for that solution we would be dividing by $0$). In general, you potentially lose solutions at points where the operation you do is invalid. Whether you actually lose solutions, and how many you lose, differs from equation to equation, and it's difficult to give a general answer.
When the thing you do could make different numbers equal, you may introduce extra solutions. Example: $$ 2\sqrt x=x-1\\ 4x=(x-1)^2\\ 4x=x^2-4x+1 $$ which has solutions $x=3-2\sqrt2\approx 0.17$ and $x=3+2\sqrt2\approx 5.83$. Note that if you insert these two solutions into the original equation, only one of them is a solution. The other one makes the left-hand side positive and the right-hand side negative, which is not equal. However, after squaring them they become equal and thus an extra solution is born. You get extra solutions at points where the two sides are different numbers, but the thing you do makes them equal. How many times this happens (and this how many extra solutions you get) differs from equation to equation, and it's difficult to give a general answer.
You also potentially get extra solutions when your manipulations give you expressions that are valid at points where your original expressions weren't valid, and similarly you potentially lose solutions when your new expressions are more restrictive.
I don't think there is a specific theorem for this. It's just something that you learn to keep in mind as you go.