Ray's New Elementary Algebra · Arts. 206–211 · Unit 7: Chapter VII: Quadratic equations
45. Pure quadratic equations
Goal: Your student solves equations with only x² and numbers, gives both roots, and solves word problems that lead to them.
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
Quadratic equations defined, pure and affected; pure quadratics solved, why there are two roots of opposite sign, exercises, and twelve problems. Answers are printed.
The lesson, step by step
Warm-up
Ask: What numbers squared give 16? (4 and −4.)
Definitions
Read Arts. 206 and 207.
A Pure Quadratic Equation is one that contains the second power only of the unknown quantity, and known terms
An Affected Quadratic Equation is one that contains both the first and second powers of the unknown quantity, and known terms
Your student sorts x² = 9, 3x² + 4x = 20 and 8x² − 5x² = 12.
The rule
Read Art. 209 and its examples.
Reduce the equation to the form ax²=b. Divide both sides by the coefficient of x², and extract the square root of both members.
Example 2: 5x² + 4 = 49; 5x² = 45; x² = 9; x = ±3.
Two roots
Read Art. 210.
1. Every pure quadratic equation has two roots, and only two.
2. These roots are equal, but have contrary signs.
Guided practice
Exercise 1, x² − 8 = 28 (±6), while you watch. Then the problems:
2. What number is that, of which the product of its third and fourth parts is equal to 108? Ans. 36.
6. What two numbers are to each other as 3 to 4, the difference of whose squares is 63? Ans. 9 and 12.
Independent practice
Exercises 2 to 7 of Art. 210 and problems 3, 4, 5, 7, 8 and 11 of Art. 211.
Wrap-up
Ask: In a problem about a number of yards, why do we keep only the + root?
Tip
Have your student write "±" every time they take a square root. Dropping a root is the common slip.
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 ↗.