Completing the square explained
Use this page to review the square-form method before applying it to your own quadratic in the app.
Start here if you want the short version before reading the full method.
- Completing the square rewrites a quadratic into a form like (x + a)^2 + b.
- It is useful for solving quadratics, spotting turning points, and understanding how the graph is shifted.
What this topic means and what to look for first.
This method is more structural than simple factorising because it shows how the quadratic is built around a squared bracket.
That makes it especially useful when you want to solve the equation and also understand what the graph is doing.
One reliable route through the topic.
- 1Move the constant term if you are solving an equation and want the square to stand alone cleanly.
- 2Take half the coefficient of x, then square that value.
- 3Add that square value to both sides if you are solving an equation, or add and subtract it inside the expression if you are rewriting only the expression.
- 4Rewrite the first three terms as a perfect square bracket.
- 5Simplify the remaining constant and, if solving, take square roots carefully.
Choose the route that fits the quadratic.
Choose this method when factorising is awkward and you want to see the square form clearly.
The completed square form makes the turning point and vertical shift easier to read.
See the method in action.
x^2 + 6x + 5
- Half of 6 is 3, and 3^2 is 9.
- Rewrite as x^2 + 6x + 9 - 9 + 5.
- This becomes (x + 3)^2 - 4.
x^2 - 4x + 1 = 0
- Move the constant: x^2 - 4x = -1.
- Half of -4 is -2, and (-2)^2 is 4, so add 4 to both sides.
- This gives (x - 2)^2 = 3, so x = 2 ± √3.
Things that commonly send the method off track.
- Forgetting to add and subtract the same square value.
- Using half the x coefficient incorrectly when it is negative.
- Balancing only one side of the equation when solving.
Use a short verification pass before moving on.
- Expand the completed square form and make sure it returns to the original quadratic.
- If you solved the equation, substitute the roots back into the original equation to confirm they work.
- Use the square form to identify the turning point and check whether it matches the shape you expect from the quadratic.
Try a few variations before switching to a calculator or solver tool.
- x^2 + 8x + 7 = 0
- x^2 - 10x + 13 = 0
- x^2 + 2x - 3 = 0
Want to try a similar problem yourself?
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A free account is the current follow-up route for returning to the solver beta and future guide updates as the public library grows.
Extra algebra revision resources
If you want more printed algebra practice after this page, these broader searches are a sensible next step.
Amazon
Algebra workbook and revision book search
Useful if you want more equation, factorising, and worked-example practice in one printed source.
View Algebra workbook and revision book searchAmazon
GCSE algebra practice resources search
A wider GCSE-style search if you want more mixed algebra questions beyond one online guide.
View GCSE algebra practice resources searchFound this useful?
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Short answers worth checking.
It makes turning points and solution structure easier to see.
It depends on the problem. It is often better for understanding structure, while the formula is a general solving method.
Continue with the next closely related topic.
Use the public site structure first, then switch into the solver tool only if you need a direct test.
CureMath uses artificial intelligence to suggest how a maths problem could potentially be solved. AI can make mistakes.
Check important answers independently before relying on them.