Basic functions: Polynomial functions
Solutions of simple quadratic equations
A quadratic equation has 0, 1, or 2 solutions. We give an example for each of the three cases .
Example: no solutions The equation \(x^2+1=0\) has no real solutions. Because a square is always greater than or equal to \(0\), the left-hand side is for each choice of \(x\) greater than or equal to \(1\).
Example: one solution The equation \(x^2-2x+1=0\) has exactly one solution. Because the left-hand side of the equation can be written as \((x-1)^2\) and this square can only be equal to \(0\) if \(x-1=0\), there is only one solution, namely \(x=1\).
Example: two solutions The equation \(x^2-4=0\) has two solutions, namely \(x=-2\) of \(x=2\).
The last example can also be understood as follows: the left-hand side of the equation, that is, \(x^2-4\), can be written as \((x+2)(x-2)\) and this product can only be equal to \(0\) if one of the terms in the product is equal to \(0\). So \(x+2=0\) or \(x-2=0\) , that is, \(x=-2\) or \(x=2\) .
We have applied the following general rule.
\[A\cdot B = 0\quad\text{if and only if }\quad A=0\;\;\text{or}\;\;B=0\] Here "or" must be understood such that \(A\) and \(B\) can also be both equal to \(0\).
In formal mathematical language, this rule is: \[A\cdot B = 0\quad\iff\quad A=0\;\;\lor\;\;B=0\]
When a quadratic equation is in factored form you can easily read of the solutions. We give another example.
Example: a x² +b x = 0 Let \(a\) and \(b\) be real numbers with \(a\neq0\). \[\begin{aligned} ax^2+bx=0&\implies x\cdot (ax+b)=0\\ \\ &\implies x=0\;\;\text{or}\;\;ax+b=0\\ \\&\implies x=0\;\;\text{or}\;\;x=-\frac{b}{a}\end{aligned}\]
In this section we will discuss three methods to determine the solutions of a quadratic equation:
- Completing the square
- Factorisation by inspection
- The quadratic formula
But before we are going to do this we look first at another simple quadratic equation, namely of the form \(x^2=c\) with a given number \(c\). The number of solutions of this equation depends on the sign of \(c\).
Example: x² = c
Two solutions
If \(c>0\), then the equation \[x^2=c\] has two solutions, namely\[x=\sqrt{c}\quad\vee\quad x=-\sqrt{c}\text.\] The parabola and the horizontal line \(y=c\) intersect in two points.
One solution
If \(c=0\), then the equation \[x^2=c\] has one solution, namely \[x=0\text.\]
The parabola and the horizontal line \(y=0\) have one common point, namely \((0,0)\).
No solution
If \(c=0\), then the equation \[x^2=c\] has no solution
The parabola and the horizontal line do not intersect \(y=c\)
Mathcentre video
Solving Quadratic Equations (50:19)
