Why are symbols not written in words?
"Integral From a x squared Plus b Plus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Plus One RightParenthesis Differential x IsGreater Integral From a x squared Plus b Minus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Minus One RightParenthesis Differential x Implies Integral From a x squared Minus b Plus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Plus One RightParenthesis Differential x IsGreater Integral From a x squared Plus b Minus c To Infinity of a Times LeftParenthesis x Plus LeftParenthesis x Plus c RighParenthesis Over LeftParenthesis x Minus One RightParenthesis Differential x" ?
Quiz:
Do you recognize this one ?
Summation On j From 1 Upto N Of y Index k Times the Product On k From 1 And k NotEqual to j Upto N Of x Minus x Index k Over x Index j Minus x Index k.
Consider this problem, taken from The Evolution of Algebra in Science, vol. 18, no. 452 (Oct 2, 1891) pp. 183-187 (taken from JSTOR, itself translated from work of Nesselman on a problem by Mohammed ibn Musa):
A square and ten of its roots are equal to nine and thirty units, that is, if you add ten roots to one square, the sum is equal to nine and thirty. The solution is as follows: halve the number of roots, that is, in this case, five; then multiply this by itself, and the result is five and twenty. Add this to the nine and thirty, which gives four and sixty; take the square root, or eight, and subtract from it half the number of roots, namely five, and there remains three: this is the root of the square which was required and square itself is nine.
This is how algebra used to be done; you have similar descriptions in Babylonian scribe tablets, Egyptian papyrii, Middle Age textbooks, etc.
Using symbols, the problem becomes, first, to solve $x^2+10x = 39$. The process is to complete the square: $$\begin{align*} x^2 + 10x &= 39\\ x^2 + 10x + 25 &= 64\\ (x+5)^2 &= 64\\ x+5 &= 8\\ x &= 3 \end{align*}$$ Something that is much easier to do without too much thought, and certainly much less effort, than the decription. Also, the idea of completing the square is much simpler to explain in symbols than it is to do so rhetorically.
Others have already answered on why one should use symbols. I want to add that one shouldn't overuse symbols, as people sometimes do.
With too many symbols, statements get clustered and confusing. Out of lazyness, many like to write $\exists$ quantors in the middle of a sentence. Others overuse e.g. $\land$, etc. ($\land$ is not a synonym for "and"!)
So basically, one shouldn't overuse symbols.