Systems of quadratic equations are systems in which at least one equation contains quadratic terms, such as \(x^2\), \(y^2\), or products of unknowns such as \(xy\).
Unlike linear systems, these systems describe nonlinear relationships between variables and may therefore admit different numbers of solutions: no solution at all, exactly one solution, or several real solutions.
Solving them generally requires a combination of algebraic techniques and geometric insight. In particular, it is essential to be able to:
- express one variable in terms of the other;
- substitute correctly into the equations;
- solve quadratic equations;
- apply standard algebraic identities;
- interpret the system geometrically.
From a geometric standpoint, solving a system means finding the points common to all the curves defined by the equations in the system.
Contents
- Definition of a system of quadratic equations
- Geometric interpretation
- Substitution method
- Method of comparison
- Systems involving circles
- Symmetric systems
- Using algebraic identities
- Number of solutions
- Checking solutions
- Common mistakes
Definition of a system of quadratic equations
A system is called a system of quadratic equations when at least one of its equations contains terms of degree \(2\).
Some examples are:
\[ \begin{cases} y=x^2,\\ y=2x+1, \end{cases} \]
or:
\[ \begin{cases} x^2+y^2=25,\\ x+y=7, \end{cases} \]
or again:
\[ \begin{cases} x^2-y^2=5,\\ xy=6. \end{cases} \]
The unknowns are generally two, denoted \(x\) and \(y\), though the method extends naturally to systems with more variables.
An ordered pair \((x,y)\) is a solution to the system if it satisfies all the equations simultaneously.
Geometric interpretation
Each equation in the system represents a curve in the Cartesian plane.
For instance:
- a linear equation represents a straight line;
- an equation of the form \(y=ax^2+bx+c\) represents a parabola;
- an equation of the form \(x^2+y^2=r^2\) represents a circle.
Solving a system is therefore equivalent to finding the points of intersection of the curves corresponding to each equation.
Consider, for example:
\[ \begin{cases} y=x^2,\\ y=x+2. \end{cases} \]
The first equation represents a parabola, while the second represents a straight line.
The solutions of the system correspond to the points where the line meets the parabola.
Geometrically, several cases may arise:
- no point of intersection;
- exactly one point of intersection;
- two or more points of intersection.
Substitution method
The most important technique for solving a system of quadratic equations is the substitution method.
The idea is to:
- express one variable from one of the equations;
- substitute that expression into the other equation;
- obtain an equation in a single unknown;
- solve the resulting equation;
- determine the value of the remaining unknown.
Let us work through a complete example.
Solve:
\[ \begin{cases} y=x^2,\\ y=x+2. \end{cases} \]
Since both equations express \(y\) explicitly, we equate the right-hand sides:
\[ x^2=x+2. \]
Bringing all terms to one side:
\[ x^2-x-2=0. \]
Factoring:
\[ x^2-x-2=(x-2)(x+1). \]
Therefore:
\[ x=2 \qquad \text{or} \qquad x=-1. \]
We now find the corresponding values of \(y\) using:
\[ y=x+2. \]
If \(x=2\), then:
\[ y=4. \]
If \(x=-1\), then:
\[ y=1. \]
Hence the solution set is:
\[ S=\{(-1,1),(2,4)\}. \]
Method of comparison
When both equations express the same variable explicitly, it is often convenient to use the Method of comparison.
Consider:
\[ \begin{cases} y=x^2-1,\\ y=2x+2. \end{cases} \]
Since both expressions are equal to \(y\), we may write directly:
\[ x^2-1=2x+2. \]
This yields a quadratic equation in the single unknown \(x\).
In practice, the method of comparison is a special case of the substitution method.
Systems involving circles
Many systems of quadratic equations involve circles.
The equation:
\[ x^2+y^2=r^2 \]
represents a circle centred at the origin with radius \(r\).
For example:
\[ x^2+y^2=25 \]
represents a circle of radius \(5\).
If the system also contains a linear equation, such as:
\[ \begin{cases} x^2+y^2=25,\\ x+y=7, \end{cases} \]
then the solutions of the system correspond to the points of intersection of the line and the circle.
Solving for:
\[ y=7-x \]
and substituting into the first equation gives:
\[ x^2+(7-x)^2=25. \]
The system is thus reduced to solving a single quadratic equation.
Symmetric systems
Some systems are called symmetric because they contain expressions that are unchanged when \(x\) and \(y\) are swapped.
Examples of such expressions are:
\[ x+y, \qquad xy, \qquad x^2+y^2. \]
Consider the system:
\[ \begin{cases} x^2+y^2=5,\\ xy=2. \end{cases} \]
In these cases it is often fruitful to exploit standard algebraic identities.
Using algebraic identities
A fundamental identity is:
\[ (x+y)^2=x^2+2xy+y^2. \]
Applying it to the system above, we obtain:
\[ (x+y)^2=5+2\cdot 2=9. \]
Therefore:
\[ x+y=3 \qquad \text{or} \qquad x+y=-3. \]
The system is thereby reduced to simpler systems.
In other problems the following identities may also prove useful:
\[ (x-y)^2=x^2-2xy+y^2, \]
or:
\[ x^2-y^2=(x-y)(x+y). \]
Recognizing these patterns often leads to significant simplifications.
Number of solutions
A system of quadratic equations may have:
- no solution;
- exactly one solution;
- two solutions;
- four solutions.
For example:
\[ \begin{cases} x^2+y^2=1,\\ x+y=3 \end{cases} \]
has no real solutions.
Indeed, substitution leads to a quadratic equation whose discriminant is negative.
Geometrically, this means that the line does not intersect the circle.
Checking solutions
In systems of quadratic equations it is essential to verify every solution found.
Checking consists of substituting each ordered pair back into the original equations of the system.
Consider, for example, the pair:
\[ (3,4) \]
in the system:
\[ \begin{cases} x^2+y^2=25,\\ x+y=7. \end{cases} \]
We verify:
\[ 3^2+4^2=9+16=25, \]
and:
\[ 3+4=7. \]
The pair satisfies both equations and is therefore a genuine solution of the system.
Common mistakes
The most frequent errors when solving systems of quadratic equations include:
- sign errors during substitution;
- incorrect expansion of binomial squares;
- overlooking some of the solutions;
- failing to check the solution pairs obtained;
- errors in factoring trinomials.
It is therefore important to proceed systematically, writing out all key steps and avoiding mental shortcuts.
In systems of quadratic equations, even a minor algebraic slip can completely invalidate the final result.