If a quadratic equation can have less then two solutions

Question

is there anyway that a quadratic equation has less than two solutions? If the first coefficient a is 0, then it is not a quadratic.

Answer 1

That depends on whether you define a double root as two solutions, and whether you count complex solutions to real quadratics. How many solutions does $x^2+2x+1=(x+1)^2=0$ have? How many solutions does $x^2+1=0$ have? If you answer two to both questions, then every quadratic has two solutions.

Answer 2

No, due to the Fundamental Theorem of Algebra, we are guaranteed for any polynomial of degree $n$ that there are $n$ roots, counted with multiplicity. So for any quadratic, $n=2$ so there are always 2 roots, but they may be the same root. That is, it may be one root with multiplicity 2, like $(x-2)^2 = 0$. This is true for any degree polynomials, so it's a good rule of thumb to know. The sum of the exponents in the factorization will always add up to the degree $n$.

Answer 3

the solution set to any equation depends on the solution space allowed. $$ X+1 = 0 $$ has no solution in $\mathbb{N}$ but can be solved in $\mathbb{Z}$ $$ X^2+1 = 0 $$ cannot be solved in $\mathbb{R}$ but has two roots in $\mathbb{C}$. astonishingly, it has an infinite set of solutions in $\mathbb{H}$, the division ring of quaternions.

the process of extending a solution space is one of the absolutely fundamental operations in mathematics. one simple procedure of this kind is the extension of an integral domain to its field of fractions.

a remarkable example, itself of great importance, is the theory of fields, where the notion of extending a solution space is perhaps the central idea.

perhaps more fancifully, you might view the completion of a metric space to be analogous to extending a solution space, if you regard a Cauchy sequence $\{a_n\}$ as an equation which may or may not have a "solution" $a_\infty$

these ideas are not unrelated, because it is one of the rather interesting basic facts of mathematics that a purely topological completion of the rationals $\mathbb{Q}$ - in the familiar Euclidean metric topology - is a sufficient enlargement to allow the solution of many irreducible polynomial equations of degree greater than one.

differential equations are another region where extension of the solution space plays a key role. the move beyond pointwise functions to generalized functions (distributions) has opened up many research areas.

as a final example we may look at group theory where an "equation" might take the form of an exact sequence: $$ 0 \to X \to F \to G \to 0 $$

a simple and important case is the presentation of a group as the quotient of a free group by a free subgroup.