Completing the Square to Classify Conics

Turn a sprawling equation into a readable one.

12 problems · about 21 minutes · TEKS P.3.B

NameDate

Figure — use these to answer the problems

Completing the square is what moves a parabola from expanded form into vertex form.

y = 1x² − 6x + 5

b-6

Figure 1

Straightforward practice. Get the method working first.

5 problems

Complete the square on x² + 6x. What is added?
Answer
2
Equal coefficients on x² and y², same sign. Which conic?
- A circle.
- A hyperbola.
Complete the square on x² + 10x. What is added?
Answer
Complete the square on y² − 12y. What is added?
Answer
(x + 5)² + (y − 1)² = 9. What is the x-coordinate of the centre?
Answer

The same ideas with more to keep track of.

3 problems

3x² + 3y² = 12. Conic type? 1 circle, 2 ellipse, 3 parabola, 4 hyperbola.
Answer
4x² − y² = 8. Conic type? 1, 2, 3 or 4.
Answer
x² + 4y² = 16. Conic type? 1, 2, 3 or 4.
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Put the completing-the-square steps in order.
Write 1 to 4 in the boxes to put these in order.
- Group the x terms and the y terms.
- Move the constant to the right side.
- Add the square of half each linear coefficient to both sides.
- Write each group as a squared binomial.
x² + y² = 0. How many points does the graph contain?
Answer
The Completion
To complete the square on x² − 8x, what number must be added?
Answer
The Radius
(x − 3)² + (y + 2)² = 16. What is the radius?
Answer