A conic rarely arrives in standard form. x² + y² − 6x + 4y − 3 = 0 is a circle, but nothing about it says so yet.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Completing the square
Group the x terms and the y terms, then add the square of half each linear coefficient to both sides.
Keep it balanced
Whatever you add on the left must be added on the right. The commonest error is completing the square and forgetting the other side.
Classifying without finishing
Compare the coefficients of x² and y². Equal means a circle, same sign but unequal means an ellipse, opposite signs mean a hyperbola, and a missing square means a parabola.
Degenerate cases
A circle equation ending in a negative radius squared has no graph at all, and one ending in zero is a single point.
Completing the square reveals the conic
An equation like 4x² + 9y² − 8x + 36y + 4 = 0 hides its identity. Completing the square in x and in y separately converts it into standard form, where centre, axes and type are all readable.
Factor out the coefficients first
Completing the square requires a leading coefficient of 1 inside each group, so factor it out before halving and squaring. Forgetting to account for that factor when balancing the equation is the standard error.
Classify before you complete
The coefficients of x² and y² already tell you the type: equal gives a circle, same sign and unequal an ellipse, opposite signs a hyperbola, one missing a parabola. Knowing the answer in advance is a useful check.
Degenerate cases exist
Some equations produce a single point, a pair of lines, or nothing at all. They arise when the slicing plane passes exactly through the cone's apex. Getting one is not necessarily an error.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Diego Find a Centre
Diego is given x² + y² − 6x + 4y − 3 = 0.
- Step 1
He groups the terms: (x² − 6x) + (y² + 4y) = 3.
Step 4: Your Turn
Practice makes it stick.
The Completion
Problem 1 of 2
To complete the square on x² − 8x, what number must be added?
The Radius
Problem 2 of 2
(x − 3)² + (y + 2)² = 16. What is the radius?
Rewrite and Classify
1 of 8
Complete the square on x² + 10x. What is added?
2 of 8
Complete the square on y² − 12y. What is added?
3 of 8
(x + 5)² + (y − 1)² = 9. What is the x-coordinate of the centre?
4 of 8
3x² + 3y² = 12. Conic type? 1 circle, 2 ellipse, 3 parabola, 4 hyperbola.
5 of 8
4x² − y² = 8. Conic type? 1, 2, 3 or 4.
6 of 8
x² + 4y² = 16. Conic type? 1, 2, 3 or 4.
7 of 8
Put the completing-the-square steps in order.
- 1Move the constant to the right side.
- 2Add the square of half each linear coefficient to both sides.
- 3Write each group as a squared binomial.
- 4Group the x terms and the y terms.
8 of 8
x² + y² = 0. How many points does the graph contain?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Complete the square on x² + 6x. What is added?
Question 2 of 2
Equal coefficients on x² and y², same sign. Which conic?
What You Learned
- Completing the square turns a general equation into a readable standard form.
- Whatever is added on the left must be added on the right.
- The two squared coefficients classify the conic before any algebra is done.