Ray's New Elementary Algebra · Arts. 215–218 · Unit 7: Chapter VII: Quadratic equations
49. Properties of the roots
Goal: Your student uses the facts that a quadratic has two roots, that their sum and product come from the coefficients, and that some quadratics have no real roots.
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
Why every quadratic has exactly two roots, factoring with the roots, forming an equation from given roots (three exercises), the sum and product of the roots, the signs of the roots in the four forms, and when the roots are impossible.
The lesson, step by step
Warm-up
Solve x² − 5x = −6. (2 and 3.)
Roots and factors
Read Art. 215.
1. Every quadratic equation has two roots, and only two.
SaySince the roots of x² − 5x + 6 = 0 are 2 and 3, x² − 5x + 6 = (x − 2)(x − 3). This is another way to factor trinomials.
Building an equation
Work the book's example: roots 4 and −6 give (x − 4)(x + 6) = 0, or x² + 2x = 24. Then your student does:
1. Find an equation whose roots are 7 and 10.
(x² − 17x = −70.)
Sum and product
Read Art. 216. In the warm-up, 2 + 3 = 5 and 2 × 3 = 6. Check: the book's form is x² − 5x = −6, so the sum is the coefficient of x with its sign changed, and the product is the known term with its sign changed.
Signs of the roots
Skim Art. 217. Then read Art. 218 and the table: 10 split as 9 + 1, 8 + 2, ..., 5 + 5. The product is largest when the parts are equal.
When a number is divided into two equal parts, their product is greater than that of any other two parts into which it can be divided.
An impossible problem
Work the book's example: divide 12 into two parts whose product is 40. Completing the square gives x − 6 = ±√(−4), impossible, since 6 × 6 = 36 is the most possible.
Wrap-up
Your student does exercises 2 and 3 of Art. 215 and explains why 12 cannot be split into parts with product 40.
Tip
Art. 217 is long and repetitive. Reading its conclusions is enough.
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 ↗.