Cogito
Precalculus · Chapter 2 · Lesson 3
Completing the Square to Classify Conics
Turn a sprawling equation into a readable one.
12 problems · about 21 minutes · TEKS P.3.B
What this lesson teaches
The student rewrites a general second-degree equation in standard form and identifies the conic.
- 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.
Warm Up
Straightforward practice. Get the method working first.
5 problemsComplete the square on x² + 6x. What is added?
Answer 9
Why 9.
Equal coefficients on x² and y², same sign. Which conic?
Answer A circle.
Why A circle.
Complete the square on x² + 10x. What is added?
Answer 25
Why (10 ÷ 2)².
Complete the square on y² − 12y. What is added?
Answer 36
Why (−6)².
(x + 5)² + (y − 1)² = 9. What is the x-coordinate of the centre?
Answer -5
Why The sign flips.
Build It Up
The same ideas with more to keep track of.
3 problems3x² + 3y² = 12. Conic type? 1 circle, 2 ellipse, 3 parabola, 4 hyperbola.
Answer 1
Why Equal coefficients on both squares.
4x² − y² = 8. Conic type? 1, 2, 3 or 4.
Answer 4
Why Opposite signs.
x² + 4y² = 16. Conic type? 1, 2, 3 or 4.
Answer 2
Why Same sign, unequal coefficients.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the completing-the-square steps in order.
Answer 1. Group the x terms and the y terms. 2. Move the constant to the right side. 3. Add the square of half each linear coefficient to both sides. 4. Write each group as a squared binomial.
Why The grouping happens before anything is added.
x² + y² = 0. How many points does the graph contain?
Answer 1
Why Only the origin satisfies it.
The Completion: To complete the square on x² − 8x, what number must be added?
Answer 16
Why 16.
The Radius: (x − 3)² + (y + 2)² = 16. What is the radius?
Answer 4
Why 4.