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I'm studying numerical analysis on college, and a question that has been following me (since I've started studying calculus) is why do we seek for the $x$ value where $f(x) = 0$? Can someone exemplify a real situation where this would be important? Because all I've seen are approaches to try to find this $x$ value, but I still don't understand why we are trying to find it.

Thanks in advance :)

Arnaldo
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VSMelo
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    "When does the ball hit the ground? Where does the ball hit the ground? What's the best place to put this bolt?" There are too many applications of solutions to nonlinear equations to even make a decent dent. Most of these applications are actually in several variables, but in math courses we do the 1D case first. – Ian May 30 '17 at 16:16
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    Many situations. The first time I had to do that was in a numerical physics exercise, where to find the Lagrange points of an orbit we had to find the minima of the potential, thus solve $\nabla V=0$, the solution had no closed form so there wasn't really an alternative. I'm sure there are many many other examples. – user438666 May 30 '17 at 16:17
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    You might note that any equation $A(x) = B(x)$ that you might want to solve can be put into the form $f(x) = 0$: take $f(x) = A(x) - B(x)$. – Robert Israel May 30 '17 at 16:17
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    Depends, what are you doing to require finding $f(x) = 0$? Are you solving polynomials? Are you solving differential equations? Are you finding the maximum of a function? There's multiple different reasons why you'd want this value. – Kaynex May 30 '17 at 16:18
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    Possible duplicate of Roots of functions / polynomials – Hritik Narayan May 30 '17 at 16:19
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    You might not realize it, but you are actually asking "Why do we solve equations?". As shown in the answers below, any equation (or system of equations) is of the form $f(x) = 0$ when you transform it suitably. – Ennar May 30 '17 at 16:32

6 Answers6

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As Robert Israel mentioned in the comments, you can solve $A(x)=B(x)$ by defining $f(x)=A(x)-B(x)$ and solving for $f(x)=0$.

A classic example of this would be finding the square root of a number: suppose we want to find $\sqrt5$, say. Then we're looking for $x$ where $x^2=5$. Let $f(x) = x^2-5$. Now you can use Newton's method (for example) to solve for $f(x)=0$.

Théophile
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    This example of finding the square root of a number was very useful. As I said, I saw methods to find $f(x) = 0$, but I had not seen an application for this. Thanks! – VSMelo May 31 '17 at 21:59
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Reason 1: Generalized Algebra

Note that $f(x)=g(x) \implies f(x)-g(x)=0 \implies h(x)=0$

General techniques to solving equations can be cast into the form $h(x)=0$ where we move everything to the LHS and call it "$h(x)$"

Reason 2: Factorizing

Note that, within sufficiently nice mathematical structures (like the real numbers you are probably used to), we have that $f(x)g(x)=0$ implies that either $f(x)=0$, $g(x)=0$, or both. This does not hold true if we replace $0$ with any other constant. This comes down to $0$ being the unique multiplicative identity constant, i.e. $a\cdot b = b$ only when $b=0$.

Reason 3: Finding Extrema

In Calculus you learn that we can find the minima and maxima of functions by finding points where $f'(x)=0$. Recall that a line of slope $0$ is "flat", and note that a function is not changing much where it is "flat".

Brevan Ellefsen
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Because $0$ is the most mainstream value, and it is conceptually easier to explain how to find $f(x)=0$.

Suppose that we want to find the value such that $f(x)=a$, then we create the function $g(x)=f(x)-a$ and we look for $g(x)=0$.

Part of the simplification is that instead of saying " if $g(x)<a$" we can just say: "if $f(x)$ is negative"

Asinomás
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We often want to "solve" general equations of the form

$g(x) = c \tag{1}$

for some constant $c$; that is, we want to find the values $x_0$ of $x$ such that (1) holds for $x = x_0$:

$g(x_0) = c. \tag{2}$

If we set

$f(x) = g(x) - c, \tag{3}$

we see that solving (1) is equivalent to solving

$f(x) = 0. \tag{4}$

Having a zero on the right-hand side is often convenient since it allows the formulation of certain iterative "solvers" such as Newton's method, which can be deployed to address (4). For this reason, texts often focus on (4), with no loss of generality.

Robert Lewis
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If you drop a ball off of a building, you can model the height of the ball with a quadratic equation with time inputs. Just google how long before ball hits the ground.

CopyPasteIt
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Here is an example that arises in calculus alot. If we take $f$ to be the derivative of a function $g:\mathbb{R}\to\mathbb{R}$ (assumed to be differentiable), then the points $x$ where $f(x)=0$ are places where relative minima and maxima of $g$ could occur.

Sri-Amirthan Theivendran
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