Complete Guide to Solving Quadratic Equations (Year 9-10 Maths)

Quadratic equations are an important part of high school Maths, especially as you progress in algebra. Whether you’re prepping for tests or want to improve your skills, learning different methods to solve quadratic equations will give you an advantage.

This guide covers four key methods, plus a free download of 10 practice questions to help you get better!

Table of contents

• Syllabus
• Outcomes
• What is a quadratic equation?
• Assumed knowledge
• Solving Quadratic Trinomials
  • Using PSF (product, sum, factor) & observations
  • Cross method
• Solving Quadratic Trinomials
  • Completing the Squares
  • The Quadratic Formula
• 10 sample questions – FREE download!

Get ahead in quadratic equations!

Solve, expand, and master quadratics with this Year 9 worksheet!

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In the NSW Syllabus NSW Stage 5.3

Quadratic equations are introduced in Year 9 and explored in more depth in Year 10. Building a strong understanding early will give you a head start and make advanced topics much easier.

According to the NSW Stage 5.3 syllabus, students should be able to:

• Solve different types of quadratic equations using various methods
• Expand special binomial products and factorise non-monic quadratic
• Work with complex linear equations, simple cubic equations, and simultaneous equations (one linear, one non-linear)

Outcomes

According the the syllabus, by the end of this topic you’ll be able to:

• Solve complex linear, quadratic, simple cubic and simultaneous equations, and rearrange literal equations (MA5.3-7NA).
• Solve a wide range of quadratic equations derived from various contexts (ACMNA269):
  • Solve equations of the form \( ax^2 + bx + c = 0 \) by factorisation and by ‘completing the square’.
  • Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \) to solve quadratic equations.
  • Solve a variety of quadratic equations.
  • Choose the most appropriate method to solve a particular quadratic equation (Problem Solving).

What is a quadratic equation?

A quadratic equation is any equation where the highest power of the variable (typically

$x$

) is 2.

The general form of a quadratic equation looks like this:

$$a{x}^2+bx+c=0$$

Where: 

• a = quadratic coefficient
• b = linear coefficient
• c = constant term

In simpler terms, quadratic equations involve a variable that is squared (raised to the power of 2). An example includes:

\( x^2\)
\(z^2 + 5z\)
\(3x^2 – 10x + 7 \)

Assumed knowledge

Before you tackle quadratic equations, make sure you’re familiar with these algebraic basics:

• Expanding binomial e.g. (
  $(x + a)(x + b)$ 

   

• Factorisation techniques (like differences of squares and perfect squares)

Four key methods to solve quadratic equations

Quadratic equations can be solved using 4 main methods:

1. Using PSF (product, sum, factor) & observation 
2. Cross method 
3. Completing the squares 
4. Using the quadratic formula

Solving quadratic trinomials (which can be factorised)

Monic quadratic trinomials

A monic quadratic trinomial is expressed in the form: \( ax^2+bx+c\) where the coefficient \( a=1\).

Using PSF (product, sum, factor) & observation 

In this method, you are expanding: \( (x + a)(x + b) = ax^2 + (a + b)x + ab \)

• The coefficient of
  $x$ 

  is \( (a+b)\)

• The constant term is \( ab\)

So, to factorise a monic quadratic trinomial, you need to reverse the process by finding 2 numbers whose:

1. Sum is the linear coefficient (x)
2. Product is the constant term

For example:

$$x^2-2x-8=0$$

You need to find two numbers that:

• Add up to the middle term (-2)
• Multiply to the constant term (-8 here)

The numbers are 2 and -4. So, you factorise it as:

$$(x+2)(x-4)=\; 0\mathrm{}$$ $x = -2$

So,

$x$

=−2

$x$

 or

$x$

=4

Non-monic quadratic trinomials  

A non-monic quadratic trinomial is expressed as: 

\( ax^2 + bx + c \)

where the coefficient \( a \neq 1 \).

Using the cross method

Step 1: Set up a workspace as shown:

Step 2: Choose numbers for the circles that multiply to = the quadratic term (the product should be equal to

$a \times x^2$

).

Step 3: Choose numbers for the squares that multiply to = the constant term.

Hint: set up and align the factors/numbers vertically. 

Step 4: Select the numbers so that when cross-multiplying in the indicated directions, the results: sum of coefficient = linear coefficient!

Step 5: Write the numbers from the circles and squares horizontally to get the two factors of the trinomial expression.

Example: Solve the quadratic equation \( 2x^2 + x – 3 = 0 \)

Set up the workspace and start filling in:

Align factors vertically and find:

• Product of circles = \( 2x^2 \) ⇒ quadratic term
• Product of squares = \( -3 \) ⇒ constant term

Cross multiply and check the sum of the coefficient results:

\( 3x + (-2x) = x \) (linear coefficient!)

Hint: Write factors horizontally, from the 1st and 2nd row, outlined in orange and green lines, respectively.

Hence,

\( 2x^2 + x – 3 = (2x + 3)(x – 1) \).

\( (2x + 3)(x – 1) = 0 \)

Therefore, \( x = \frac{-3}{2} \text{ or } x = 1 \)

Note: Sometimes, before factorising, you need to extract the Highest Common Factor (HCF) to simplify the quadratic trinomial.

Example:

$$6x^2+3x-9$$

The HCF is 3, so we factor it out first:

$$6x^2+3x-9=3(2x^2+x-3)$$

Then, we factorise the remaining quadratic:

$$3(2x+3)(x-1)$$

Solving Quadratic Trinomials (which cannot be factorised)

Completing the squares

Completing the square rearranges a quadratic equation into a perfect square trinomial (a squared binomial). This allows us to solve for

$x$

by square-rooting both sides.

\( (a \pm b)^2 = a^2 \pm 2ab + b^2 \)

To factorise by completing the square: 

1. Express the equation in monic form (divide through by coefficient of 𝑥²).
2. Add and subtract half of the linear coefficient squared.
3. Factorise out the perfect squares and combine the constant terms.
4. Move constant terms to the other side of the equation and solve by square-rooting both sides.
5. Make 𝑥 the subject and solve for 𝑥.
   $$ 

To complete the square for \( ax^2 + bx + c \) (where a=1):

1. Take half of the linear coefficient (the coefficient of x, which is b)
2. Square this value: \( \left( \frac{b}{2} \right)^2 \)
3. Add and Subtract \( \left( \frac{b}{2} \right)^2 \) within the equation, then write it as a perfect square trinomial and leave the constant.

\( x^2 + bx + \frac{b^2}{4} – \frac{b^2}{4} + c \) \( = \left( x + \frac{b}{2} \right)^2 – \frac{b^2}{4} + c \)

Example: Solve  \( x^2 + 6x + 1 \):

1. Half the linear coefficient (\( b=6\)) is 3
2. Squaring it: \( 3^2 = 9 \)
3. Add and subtract 9, rewrite:

\( (x + 3)^2 – 8 = 0 \)

\( (x + 3)^2 = 8 \)

\( x + 3 = \pm 2\sqrt{2} \)

\( x = \pm 2\sqrt{2} – 3 \)

Note: Ensure quadratic coefficient = 1 (monic)

To factorise non-monic, divide through by coefficient of \( x ^2 \)

Examples: \( 2x^2 – 4x = 5 \) can be rewritten as \( x^2 – 2x – \frac{5}{2} = 0 \)

The Quadratic Formula

When a quadratic equation can’t be factorised, or if completing the square seems too tricky, use the Quadratic Formula.

This formula works for all quadratic equations:

\( x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \)

Apply the formula by substituting variables: \( ax^2 + bx + c \)

Example:

Solve the quadratic equation: \( x^2 + 6x + 2 = 0 \).

1. Find values of a, b, c.

a = 1, b = 6, c = 2

2. Substitute these values into the quadratic formula.

\( x = \frac{-6 \pm \sqrt{(6)^2 – 4(1)(2)}}{2(1)} \)

\( x = \frac{-6 \pm \sqrt{36 – 8}}{2} \)

\( x = -3 \pm \sqrt{7} \)

Practise your quadratic equation skills!

Get ahead in quadratic equations!

Solve, expand, and master quadratics with this Year 9 worksheet!

Download now

Get ahead in quadratic equations!

Fill out your details below to get this resource emailed to you.

"*" indicates required fields

First Name*

Last Name*

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Grade in 2025*

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