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Completing the square rewrites a quadratic polynomial so that its minimum or maximum and its roots of a polynomial become easy to read. The linear term is absorbed into a shift of the variable; the resulting polynomial identity changes the description, rather than the values of the polynomial.

For coefficients $a,b,c$ in the real numbers, with $a\ne 0$,First factor out the leading coefficient from $a{x}^2+bx$, then add and subtract $(b/(2a){)}^2$ inside the bracket. The cross term in $(x+b/(2a){)}^2$ is $(b/a)x$, exactly the term needed. Multiplication by $a$ restores $bx$. Thus the half-coefficient is $b/(2a)$, rather than $b/2$ unless the quadratic polynomial is a monic polynomial. Expanding the right-hand side verifies the polynomial identity directly.^[ref]

For an original example,For every $x$ in the real numbers, $(x+3{)}^2\ge 0$, with equality only at $x=-3$. The minimum is therefore $4$, reached at $-3$; no calculus is needed, and there are no roots of a polynomial in the real numbers. The expanded coefficients conceal these conclusions more effectively than the completed square does.