Ray's New Elementary Algebra · Art. 212 · Unit 7: Chapter VII: Quadratic equations
46. Completing the square
Goal: Your student solves any quadratic equation by completing the square.
You'll need
- The book, open to the page
- Pencil and plenty of paper (or a whiteboard)
- A card to cover printed answers
What's in the book
Two opening examples, the four forms of affected quadratics, why adding the square of half the coefficient of x makes a perfect square, the rule, worked examples 1 to 6, and exercises 7 to 39. Answers are printed.
The lesson, step by step
Warm-up
Square x + 3 and x − 5. (x² + 6x + 9; x² − 10x + 25.) Ask: How is the last number related to the middle coefficient?
Make it a square
Read examples 1 and 2 on p. 194. In x² + 6x = 16, adding 9 makes x² + 6x + 9 = 25, so x + 3 = ±5 and x = 2 or −8.
The rule
Read the rule on pp. 195 and 196.
4. Add to each member the square of half the coefficient of x.
5. Extract the square root of both sides.
6. Transpose the known term to the second member.
Work an example together
1. Find the roots of the equation x²+8x=33.
Half of 8 is 4; 4² = 16. x² + 8x + 16 = 49. x + 4 = ±7. x = 3 or −11. Verify both.
Guided practice
7. x²+8x=20 Ans. x=2, or —10.
14. x²— 8x=— 15 Ans. x=5, or 3.
Your student works both while you watch. Then read example 5, where the equation must first be cleared of fractions.
Independent practice
Exercises 8 to 22. Exercise 17 (2x² − 5x = 12) needs step 3, dividing by the coefficient of x², first.
Check
Substitute both roots. After working by hand, your student may confirm on the site's calculator (/homeschool/calculator/) with solve(x^2+8*x=33, x).
Wrap-up
Ask your student to explain, with a drawing of a square, why half the coefficient is used.
Afterwards
Exercises 23 to 39 make a good second session, especially the literal ones (34 to 39).
Tip
A picture helps: x² + 8x is a square x by x plus two strips 4 by x. The missing corner is 4 by 4 = 16.
Answers to the book's problems
The book prints the answer after each exercise.
Worked out for this site from the scan. If a number in your copy differs, trust the book. Every answer is in the Key ↗.