Solving Quadratic Equations and Simultaneous Equations
Solving Quadratic Equations and Simultaneous Equations
Quadratic equations and simultaneous equations are important topics in Secondary Math and Additional Math. Students often meet them in algebra, graphs, coordinate geometry, word problems and exam-style application questions.
This guide explains what quadratic equations are, the main methods used to solve them, and how quadratics can appear when solving simultaneous equations.
Key idea: A quadratic equation usually contains an x2 term. When solving simultaneous equations, substituting one equation into another can often produce a quadratic equation that must be solved carefully.
What Is a Quadratic Equation?
A quadratic equation is an equation that can be written in the standard form:
The solutions of a quadratic equation are also called its roots. These are the values of x that make the equation true.
Common Methods to Solve Quadratic Equations
1. Factorisation
Factorisation is useful when the quadratic expression can be written as a product of two brackets.
(x + 3)(x − 1) = 0
So, x = −3 or x = 1.
2. Quadratic Formula
The quadratic formula can be used when factorisation is difficult.
| −b ± √(b2 − 4ac) |
| 2a |
This works for any quadratic equation in the form ax2 + bx + c = 0.
How Quadratics Appear in Simultaneous Equations
In simultaneous equations, students are often given two equations that must be solved together. When one equation is linear and the other is non-linear, substitution may produce a quadratic equation.
For example, solve:
2x2 − y2 = 2
Step 1: Make y the subject
From x + y = 1:
Step 2: Substitute into the second equation
Step 3: Expand and simplify
Step 4: Factorise
(x + 3)(x − 1) = 0
Step 5: Find y
Using y = 1 − x:
Practice Example: Rectangle Dimensions
A rectangle has a perimeter of 36 m, and the square of the length of its diagonal is 170 m2. Find the dimensions of the rectangle.
Step 1: Define the variables
Let the length and width of the rectangle be x and y.
Since the perimeter is 36 m:
x + y = 18
So:
Step 2: Use Pythagoras’ Theorem
The square of the diagonal is 170 m2, so:
Substitute y = 18 − x:
Step 3: Expand and simplify
Step 4: Factorise
(x − 7)(x − 11) = 0
Step 5: Find the dimensions
If x = 7, then y = 11.
If x = 11, then y = 7.
Common Mistakes Students Make
Wrong Expansion
Students often expand (1 − x)2 wrongly. Remember that (1 − x)2 = 1 − 2x + x2.
Sign Errors
A negative sign outside a bracket changes every term inside the bracket.
Forgetting the Second Solution
Quadratic equations may have two valid solutions. Always check both values.
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