Year 10 Maths Algebra: How to Solve Quadratic Equations [Free Algebra Worksheet]
Posted on February 18, 2017 by Matrix Education
Being able to solve quadratic equations is an essential skill necessary for a number of topics such as curve sketching, and for finding the minimum or maximum values to solve reallife problems.
NSW Syllabus Outcomes
Stage 5.3: Solve a wide range of quadratic equations derived from a variety of contexts (ACMNA269)

 solve equations of the form by factorisation and by ‘completing the square’
 use the quadratic formula to solve quadratic equations
Note To Students
This page will teach you the methods for factorising quadratic expressions of the form . For all other factorising techniques such as:
 Finding common factors
 Grouping in pairs
 A difference of two squares
 Perfect squares
Please visit Year 9 Maths Algebra – Factorisation Techniques
Assumed Knowledge
Students should be familiar with basic algebraic techniques including expanding special binomial products and simple arithmetic. Knowledge of factorisation techniques will also be required.
Content
A quadratic trinomial is an algebraic expression in the form:
Where:
 is called the quadratic coefficient or the leading coefficient
 is called the linear coefficient
 > is called the constant term
A quadratic equation of the form can have up to two real solutions. When we ‘solve’ a quadratic equation, we are looking for the values that make the equation true.
From basic arithmetic we know that if the product of the two numbers is zero, then at least one of the numbers must be zero.
 In symbols: If , then either or or both and are zero.
 For example: If , then either , or , or both are zero. Hence or .
To solve a quadratic equation, you must move all the nonzero terms onto the left hand side first so your equation is in the form . Then, try factorising the left hand side!
Example
Solve the quadratic equation .
Solution
First we must factorise the expression on the left hand side, .
Hence we are solving the quadratic equation .
It can be seen that or for the equation to be true.
Therefore the solutions are or .
The factorisation used in this example is simple as we only needed to extra a common factor to factorise. However, when you are given a quadratic trinomial to solve, you will need to use the following methods.
Solving Quadratic Trinomials (which can be factorised)
1. Monic Quadratic Trinomials
A monic quadratic trinomial is an expression of the form where .
When we expand , we get .
The coefficient of is and the constant term is .
Hence, to factorise a monic quadratic trinomial, we must reverse the process by finding two numbers whose:
 Sum is the linear coefficien
 Product is the constant term
Example: Solving Monic Quadratic Equation
Solve the quadratic equation .
Solution
Factorising the expression , we must find two numbers whose sum is and whose product is . The only possible numbers are and .
Therefore
Hence we are solving the equation .
It can be seen that or .
Therefore the solutions are or .
2. Nonmonic Quadratic Trinomial
A nonmonic quadratic trinomial is an expression of the form where . There are three main strategies for factorising these types of expressions.
(a) Pairing Method
1. Find two numbers and whose sum is the linear coefficient and whose product is the product of the quadratic coefficient and the constant.
2. Use the two numbers to split the middle term into the form .
3. Complete the factorisation by grouping in pairs.
Note To Students
You may revise the ‘grouping in pairs’ technique by visiting Year 9 Maths Algebra – Factorisation Techniques.
Example: Solving Nonmonic Quadratic Equation
Solve quadratic equation .
Solution
To factorise the expression , we first find the product of the quadratic coefficient and the constant, . Now, we must find two numbers whose sum is and whose product is . The only possible numbers are and .
We use these two numbers to split the middle term and then factorise by grouping in pairs.
Hence, we are solving the equation .
Therefore the solutions are or .
(b) Fraction Method
1. Rewrite the trinomial expression as the fraction in the form .
2. Find two numbers whose sum is the linear coefficient and whose product is the product of the quadratic coefficient and the constant.
3. Place these factors into the empty brackets.
4. Factorise each bracket where possible by extracting common factors.
5. Cancel out common factors.
Example: Solving Nonmonic Quadratic Equation using Fraction Method
Solve the quadratic equation .
Solution
To factorise the expression , rewrite the trinomial as a fraction.
Find two numbers whose sum is and whose product is .
The only possible numbers are and .
Place these factors into the brackets, factorise and cancel.
Hence, we are solving the equation .
Therefore the solutions are or .
(c) Cross Method
Step 1: Set up a workspace as shown:
Step 2: Place numbers in the circles such that the product of the circles is the quadratic term.
Step 3: Place numbers in the squares such that the product of the squares is the constant term.
Step 4: The numbers should be chosen such that cross multiplying in the direction of the arrows and adding the two results will give the linear coefficient.
Step 5: Read across each row and combine the circle and the square the find the two factors of the trinomial expression.
Example: Solving Nonmonic Quadratic Equation using Cross Method
Solve the quadratic equation .
Solution
First we must factorise the expression .
Setting up the workspace and filling in the circles and squares, we get:
The product of the expressions in the circles , the quadratic term. The product of the numbers in the squares give , the constant term. Crossmultiplying, we can see that the sum of the results is , the linear coefficient.
Hence, .
We are solving the equation
Therefore the solutions are or .
Note To Students
Sometimes it may be necessary to extract a highest common factor (HCF) from the expression before factorising the quadratic trinomial using these strategies. For example, the expression can be factorised by first removing the HCF of 3.
e.g. .
Solving Quadratic Trinomials (which cannot be factorised)
When a quadratic expression cannot be factorised to give rational solutions as we have seen in the examples so far, we need to find other ways to solve the quadratic equation.
1. Completing the Square
This technique uses the special binomial products referred to as perfect squares:
To factorise by completing the square:
 Express the quadratic equation in the form where and are real numbers.
 Complete the square on the left hand side of the equation by halving the linear coefficient, squaring it, and adding it to both sides of the equation.
Example: Solving Quadratic Equations using Completing the Square Method
Solve the quadratic equation .
Soution
Move the constant over to the right hand side.
Complete the square on the left hand side by halving the linear coefficient, squaring it and adding it to both sides of the equation.
Express the left hand side as a perfect square and simplify the right hand side.
Solve the resulting equation by taking the square root of both sides of the equation.
Therefore, the solutions are
Note To Students
This method of completing the square only works on monic quadratic expressions i.e. quadratic coefficient . To factorise a nonmonic quadratic using this method, start by dividing through by the coefficient of .
For example, can be rewritten as . The method of completing the square can now be applied to this equation.
2. The Quadratic Formula
Applying the method of completing the square to the general quadratic equation , a general formula for the solutions of the equation (if they exist) can be derived. This is called the quadratic formula, given by:
Note To Students
At Matrix, we will provide you with an extensive proof for the derivation of this formula.
Example: Solving Nonmonic Quadratic Equation using The Quadratic Formula
Solve the quadratic equation .
Solution
Find the values of , and .
Substitute these values into the quadratic formula.
Therefore, the solutions are .
Note To Students
The quadratic formula is able to solve any quadratic equation!
Summary
To solve a quadratic equation:
1. Extract the highest common factor (if any)
2. Factorise the quadratic expression if possible. If it is a nonmonic quadratic trinomial, use one of the following factorisation methods:
(a) Pairing method
(b) Fraction method
(c) Cross method
3. If it is not possible to factorise the quadratic expression, use one of the following methods:
(a) Completing the square
(b) Quadratic formula
Year 10 Algebra Worksheet – Solving Quadratic Equations
Check your skills with the following 10 exercises!
1.
2.
3.
4.
5.
6.
7. (use completing the square method)
8. (use completing the square method)
9. (use the quadratic formula)
10. (use the quadratic formula)
Solutions
1. or
2. or
3. or
4. or
5. or
6. or
7.
8.
9.
10.
Want to take your Year 10 Maths skills next level?

 Need more practice with factorising quadratic expressions? Try our Year 9 Maths Max Series Volume 1: An Exam Preparation Workbook that contains examples and questions on the topics ‘Algebraic Techniques and Surds & Indices’.
 Read our list of 5 proven and effective maths study tips!
 Gain an indepth knowledge & understanding and the problem solving skills required to ace your Maths exam at our Year 10 Maths Advanced Courses.
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