Here we are going to explain everything what you must need to know about quadratic simultaneous equation for GCSE maths ( Edexcel ,AQA , OCR ) .
Before we begin with the concept let's discuss some basic terms that I hope you are already familiar with. so let's quickly recall them in sequence
Linear equation : Any equation that express a relationship between two or more variables of degree one provided that the variables are not multiplied together. Let's see some examples of linear equations.
Linear equation in one variable : A linear equation having one variable
Ax +b = 0 (Where A \( \ne 0 \) )
3x +8 =0 ,
2x -4 = 0
Linear equation in two variables : A linear equation having two variables
Ax + By = C (General form)
2x +3y = 6 ,
y = 4x + 3
Imp : Every linear equation is a polynomial of degree one and the graph of a Linear equation in one or two variables always result in a straight line.
Quadratic equation : A polynomial of the form \( ax^2 +bx +c =0 \) provided that a \( \ne 0\) .If a =0 then it will convert to linear equation. Look at the examples below .
\( 2x^2 -4x + 3 =0 \),
\( 5x^2 -x + 10 =0 \) ,
\( y = x^2 -4 \)
Imp: The graph of a quadratic equation results in a parabolic shape.
What are Quadratic Simultaneous Equations
A system of two or more equations where at least one of the equations is a quadratic equation (which means the highest power of the variable is eihter \( x^2 \) or \( y^2 \) ) called Quadractic Simultaneous equation.
Below is given some examples of quadratic simultaneous equations. Each system containes one linear equatrion and one quadratic equation or both quadratic equations.
Quadratic simultaneous equations can have zero, muliple or infinitely many solutions depending upon the nature of the curve for each equation.
No solution - Curves don't intersect at any point.
One Solution - Curves intersect at one point only.
Multiple solutions - Curves intersect two or more than two points.
Infinite solutions - Curves are identical.
Example -1:
\( y = x + 1 \)
\( y = x^2 - 3x + 1 \)
Example -3 :
\( y = x^2 \)
\( y = -x^2 + 10x - 12 \)
How to Solve Quadratic Simultaneous Equation
To solve quadratic simultaneous equations we use substitution method followed by factorization method or quadratic formula to solve the resulting quadratic equation.
If the roots of a quadratic equations are imaginary then no real solution exists for the system of Quadratic simultaneous equations.
Let's Solve the above given examples of quadratic Simultaneous Equations one by one.
1. \( x^2 + y^2 = 5^2\) ,
\( y = x + 1 \)
Ans: Substitution method: Put \( y = x + 1 \) in \( x^2 + y^2 = 5^2\)
\( x^2+(x+1)^2 = 25 \)
\( x^2+x^2+2x+1=25 \)
\( 2x^2+2x−24=0 \)
\( x^2+x−12=0 \)
(x+4)(x−3)=0
Therefore,
x=−4 or x=3
Using y=x+1:
If x=−4, then y=−3
If x=3, then y=4
The solution pair for the system is (−4,−3) and (3,4).
2. \( y = x + 1 \)
\( y = x^2 - 3x + 1 \)
Ans: Substitution method:Equate both the equations
\( x + 1 = x^2 - 3x + 1 \)
\( 0 = x^2 - 4x \)
\( x^2 - 4x = 0 \)
Now Factorise:
\( x(x - 4) = 0 \)
Therefore,
\( x = 0 \) or \( x = 4 \)
Using \( y = x + 1 \):
If \( x = 0 \), then \( y = 1 \).
If \( x = 4 \), then \( y = 5 \).
The solution pair for the system is (0,1)) and (4,5).
3. \( y = x^2 \)
\( y = -x^2 + 10x - 12 \)
Ans: Substitution Method : Equate both equations
\( x^2 = -x^2 + 10x - 12 \)
\( 2x^2 - 10x + 12 = 0 \)
\( x^2 - 5x + 6 = 0 \)
Now Factorise:
\( (x - 2)(x - 3) = 0 \)
Therefore,
\( x = 2 \) or \( x = 3 \)
Now find \( y \) using \( y = x^2 \):
If \( x = 2 \), then \( y = 4 \)
If \( x = 3 \), then \( y = 9 \)
The solution pair for the system is (2,4) and (3,9).
