MHB Why can't real numbers satisfy this absolute value equation?

AI Thread Summary
The absolute value equation |x^2 + 4x| = -12 has no solutions in real numbers because the absolute value is always non-negative, meaning it cannot equal a negative number. The lowest value of |x^2 + 4x| occurs at its vertex, which is also non-negative. For complex numbers, the absolute value is defined as the square root of the sum of the squares of the real and imaginary parts, which is also non-negative. Therefore, there are no real or complex solutions to the equation. The discussion emphasizes the fundamental property of absolute values being non-negative.
mathdad
Messages
1,280
Reaction score
0
Explain, in your own words, why there are no real numbers that satisfy the absolute value equation | x^2 + 4x | = - 12.

Can we say there is no real number solution here? If so, is the answer then imaginary taught in some advanced math class?
 
Mathematics news on Phys.org
There's no solution, real or otherwise. What's the lowest value $|x^2+4x|$ can have? After answering that, consider the RHS of the equation.
 
For x a real number, |x| is defined as "x is x is non-negative, -x if x is negative". From that it should be clear that |x| is never negative.

For x a complex number, written as a+ bi, |x| is $$\sqrt{a^2+ b^2}$$. That is also never negative.
 
Thank you. Someone told me that there complex solutions but that did not make sense. So, I decided to ask the real math guys. Thank you again.
 

Thread 'Non-orthogonal bases'

Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
View full post »

Thread 'Fermat's Last Theorem'

Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
View full post »

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Similar threads

MHB Is it Possible for an Absolute Value Equation to Equal a Negative Number?
Replies
4
Views
2K
MHB Why Are There No Real Solutions to the Equation \( |x^2 + 4x| = -12 \)?
Replies
4
Views
2K
MHB Absolute Value Equation |3x - 2|/|2x - 3| = 2
Replies
10
Views
2K
MHB Proving an absolute value inequality
Replies
2
Views
3K
B Is 0 a Real Number, an Imaginary Number, or Both?
Replies
36
Views
6K
MHB Min(a, b) expressed in terms of absolute value
Replies
17
Views
27K
MHB Solve Real Number $x$ That Satisfies Equation
Replies
4
Views
1K
B Why are the values (-1.618, 0) and (0.618 ,0) solutions for this equation?
Replies
44
Views
5K
When to introduce absolute value in hyperbola expression?
Replies
7
Views
2K
Solving Absolute Value Problem: x ≤ 3
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top