Completing Square Method Worksheet & Examples (Step-by-step)

The solution of quadratic equations represents a crucial mathematical skill that IGCSE Maths students must master. One of the multiple solving methods that includes factoring and using the quadratic formula or graphical solutions, the completing square method performs best because it directly connects to the algebraic foundations related to geometry. Using this method, you can transform any quadratic equation into basic perfect square trinomials to simplify your solution steps.

During the 9th century, Muhammad ibn Musa al-Khwarizmi published “Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala” to introduce algebraic rules and square completion methods used to solve quadratic equations. Through his research, he developed concepts that eventually resulted in the term “algebra.” The European thinker René Descartes expanded previous progress through the creation of symbolic notation, which unified algebra with geometry. The “Completing Square Method” remains an integral component that students must master for IGCSE Mathematics today.

In this blog post, you’ll find:

A full breakdown of the completing square method
5 solved examples, from basic to advanced
A downloadable worksheet of 20 practice problems
Answers (not solutions) for self-evaluation

What is the Completing Square Method?

To “complete the square” means to rewrite a quadratic expression of the form:

ax² + bx + c

into a perfect square trinomial that looks like:

a(x+d)² + e

This process simplifies the equation, allowing us to solve it by taking square roots on both sides.

✅ Step-by-Step Guide

To solve a quadratic equation using completing the square:

Move the constant to the other side.
Divide all terms by the coefficient of x² (if it’s not 1).
Add the square of half the coefficient of “x” to both sides.
Factor the left-hand side as a perfect square.
Solve by taking the square root.
Isolate “x”.

🧠 5 Solved Examples Of Completing Square Method (From Easy to Advanced)

Example 1: x²+6x+5=0

Step 1: Move constant \[ \\ x^2 + 6x = -5 \\ \] Step 2: Add \( \left( \frac{6}{2} \right)^2 = 9 \) to both sides ; \[ \\ x^2 + 6x + 9 = -5 + 9 \\ \] Step 3: \[ (x + 3)^2 = 4 \] Step 4: \[x + 3 = \pm 2 \] Answers: \[x = -1 \space\space or \space\space x = -5 \]

Example 2: x²-4x+1=0

Step 1: \[ \\ x^2 – 4x = -1 \\ \] Step 2: Add \( \left( \frac{-4}{2} \right)^2 = 4 \) to both sides ; \[ \\ x^2 – 4x + 4 = -1 + 4 \\ \] Step 3: \[ (x – 2)^2 = 3 \] Step 4: \[ x – 2 = \pm \sqrt{3} \] Answers: \[ x = 2 + \sqrt{3} \space\space or \space\space x = 2 – \sqrt{3} \]

Example 3: 2x²+8x-3=0

Step 1: Divide by 2 \[ x^2 + 4x = \frac{3}{2} \] Step 2: Add \(\left(\frac{4}{2}\right)^2 = 4\) \[ x^2 + 4x + 4 = \frac{3}{2} + 4 = \frac{11}{2} \] Step 3: \[ (x + 2)^2 = \frac{11}{2} \] Step 4: \[ x = -2 \pm \sqrt{\frac{11}{2}} \]

Example 4: 3x² – 12x + 7 = 0

Step 1: Divide by 3 \[ x^2 – 4x = -\frac{7}{3} \] Step 2: Add \(\left(\frac{-4}{2}\right)^2 = 4\) \[ x^2 – 4x + 4 = -\frac{7}{3} + 4 = \frac{5}{3} \] Step 3: \[ (x – 2)^2 = \frac{5}{3} \] Step 4: \[ x = 2 \pm \sqrt{\frac{5}{3}} \]

Example 5 (Advanced): x² + 1.5x − 1.25 = 0

Step 1: \[ x^2 + \frac{3}{2}x = \frac{5}{4} \] Step 2: Add \(\left( \frac{3}{4} \right)^2 = \frac{9}{16}\) \[\\LHS: \space\space x^2 + \frac{3}{2}x + \frac{9}{16} = \frac{5}{4} + \frac{9}{16} \\ RHS: \frac{20}{16} + \frac{9}{16} = \frac{29}{16} \] Step 3: \[ \left( x + \frac{3}{4} \right)^2 = \frac{29}{16} \] Step 4: \[ x = -\frac{3}{4} \pm \sqrt{\frac{29}{16}} = -\frac{3}{4} \pm \frac{\sqrt{29}}{4} \]

📥 Completing Square Method Downloadable Practice Worksheet (with Answers)

We’ve created a comprehensive worksheet with 20 practice questions on solving quadratic equations using the completing square method. These cover:

Simple quadratics
Quadratics with leading coefficient ≠ 1
Rational coefficients
Surds
Negative discriminants

✨ You’ll find all the answers at the end of the worksheet—perfect for self-checking!

[Click here to download the Completing the Square Worksheet (PDF)]

💬 Conclusion: Learn Math Confidently with My Maths Club

We hope this complete guide helped you master the completing square method—from theory to practical application. Whether you’re tackling standard equations or dealing with advanced surds and fractions, this technique sharpens your algebraic understanding and prepares you for exam success.

If you’re looking for live, interactive math tuition that simplifies even the toughest concepts, then join our IGCSE Maths classes at MyMathsClub.com.
📚 Enjoy: