The notion of a quadratic equation in one unknown

Basic functions: Polynomial functions

The notion of a quadratic equation in one unknown

Solving a quadratic equation in one unknown by reduction Suppose that \(x\) symbolizes the number \(3\), then, for example, \(x^2-2x=4x-9\) is a valid statement. This means that \(x=3\) satisfies the equation \(x^2-2x=4x-9\). We call this a quadratic equation because it contains the quadratic term \(x^2\). For the number 3 you can write many more equations that are satisfied.

In practice, the situation is the other way around: then \(x\) is an unknown number that satisfy the equation \(x^2-2x=4x-9\) and you are trying to find the possible values of \(x\) \(x\). In other words, you want to solve the equation. This is possible by reduction to a basic form, i.e. by repeatedly writing an equivalent equation that is simpler than the previous one but has the same solution until you arrive at a quadratic equation in basic form. In the given example, it may go as follows:

Solving the equation \(x^2-2x=4x-9\).

Subtract from the left- and right-hand side \(4x\): \(\;x^2-2x-4x=4x-9-4x\),
which simplifies to \(\;x^2-6x=-9\).
Add \(9\) to the left- and right-hand side: \(\; \;x^2-6x+9=-9+9\),
which simplifies to \(\;x^2-6x+9=0\).
The quadratic equation is now in a basic form
Recognise the left-hand size as a special product: \(\;(x-3)^2=0\)..

So, the solution is \(x=3\).

The steps in solving by reduction to a basic form are:

adding or subtracting the same term on both sides of the equation;
multiplying or dividing both sides by a nonzero number;
collecting similar terms.

It may happen that you finally arrive at a quadratic equation in basic form for which there are no real solutions. This means that the original equation and all equivalent intermediate equations have no real solution.

Example with no real solution Suppose that you try to solve the equation\(x^2+3x+1=x-1\). After subtraction of \(x\) from the left- and right-hand side, you get the equation \(x^2+2x+1=-1\) that can be rewritten as \((x+1)^2=-1\), Because a square of a real number cannot be negative, the quadratic equation has no real solution. In this section we will always assume the we work only with real numbers for solving a quadratic equation.

We end this theory page with common terminology that we will use henceforth.

General terminology Let \(x\) be a variable.

A quadratic equation with unknown \(x\) is an equation that can be reduced, by elementary operations, to a basic form \[ax^2+ bx+c = 0\] where \(a, b\) and \(c\) are numbers and \(a\neq 0\).

There is no unique basic form: the equations \(2x^2+2x+2=0\) and \(x^2+x+1=0\) are both in the basic form, but are different and can be carried over into one another through elementary operations.

With an elementary operation we mean expansion of brackets, the regrouping of subexpressions, the addition or subtraction of the same expression on either side of the equation, or the multiplication or division by a nonzero number on both sides of the equation. We speak of a elementary reduction when all the steps in the reduction are elementary operations.

The expression to the left of the equal sign ( \(=\) ) is called the left-hand side of the equation (above that \(ax+ by + c\) ) and the expression on its right is called the right-hand side (above this is \(0\) ).

The expression \(ax^2\), \(b x\) and \(c\) in the left-hand side of the basic form are called terms. The number \(a\) is the coefficient of \(x^2\) and we call \(ax^2\) the quadratic term in the equation. The number \(b\) is the coefficient of \(x\) and we call \(bx\) the linear term in the equation. Terms that do not contain an unknown are called constant terms, or constants for short (above, these are the numbers \(c\) and \(0\)).

A number \(s\) is called a solution of the equation if substitution of \(x=s\) turns the equation into a true statement. All values of \(x\) for which the equation is true constitute the solution of the equation.

Two quadratic equations are called equivalent when they have the same solutions because they can be transformed into one another by elementary reduction.
To indicate that two equations are equivalent, the symbol \(\Leftrightarrow\) can be used; for example \(8x^2=2\Leftrightarrow 4x^2=1\) and \(4x^2=1\Leftrightarrow (2x)^2=1\)..

If two equations can be reduced to the same basic form, then the two equations are equivalent.

Although the equations \(\;\frac{x+2}{x-1}=x+1\;\) and \(\;|x^2-1|=|-x^2-2x|\;\) can be solved using quadratic equations (check this yourself), we still cannot call them quadratic equations, because the left-hand side of the first equation is a rational expression and the second equation contains absolute values. As a result, there is no elementary reduction to the basic form of a quadratic equation.

The equations \((x^2-1)^2=(x^2+1)^2\) and \((x^2+2)^2=(x^2+1)^2\) are considered quadratic, because expanding the brackets and regrouping terms lead to the equations \(4x^2=0\) and \(2x^2+3=0\). It would be too artificial to consider these equations as polynomial equations of degree 4, just because there is a square of a square in the original equation.

Conversely, we call the equation \(\sqrt{(x^2-1)^2+4x^2}=0\) not quadratic, even though it can be reduced to \(x^2+1=0\). However, the reduction does contain a non-elementary step: the transition from \(\sqrt{(x^2+1)^2}=0\) to \(x^2+1=0\).

The concept of equivalence of equations may also be used for nonquadratic equations:

Two equations are equivalent if they have the same solution(s).

Each of the two nonquadratic equations \((x^2-1)^3=0\) and \((x^2-1)^2=0\) is equivalent to the quadratic equation \(x^2-1=0\). However, the first two equations are nonquadratic because, in order to reduce to the quadratic equation \(x^2-1=0\), a non-elementary simplification is needed. It are examples of polynomial equations of degree 6 and 4, respectively.

Non-elementary operation

Squaring both sides of an equation is an example of an operation that is not elementary and that does not always transform an equation into an equivalent one. For instance, squaring both sides of the equation \(x=5\) gives \(x^2=25\), an equation having \(x=-5\) as a solution next to \(x=5\). Besides, this operation transforms many linear equations into non-linear ones.