Concept Byte: Writing Equations From Roots

For Use With Lesson 4-5

The root of an equation is a value that makes the equation true. You can use the Zero-Product Property to write a quadratic function from its zeros or a quadratic equation from its roots.

Activity 1

1. 1. Write a nonzero linear function f (x) that has a zero at x = 3.
   2. Write a nonzero linear function g(x) that has a zero at x = 4.
2. 1. For f and g from Exercise 1, write the product function h ( x ) = f ( x ) · g ( x ) . h open x close equals f open x close middle dot g open x close .
   2. What kind of function is h(x)?
   3. Solve the equation h(x) = 0.

Mental Math Write a quadratic equation with each pair of values as roots.

3. 5 and 3
4. 2.5 and 4
5. − 4 negative 4  and 4
6. 5 and 10
7. 3 2 3 halves  and − 2 negative 2

You can also use zeros or roots to write quadratic expressions in standard form.

Activity 2

8. 1. Copy and complete the table. Write the product ( x − a ) ( x − b ) open x minus eh close open x minus b close  in standard form for each pair a and b.
   2. Is there a pattern in the table? Explain.
9. 1. If you know the roots, you can write a quadratic function or equation in standard form. Explain how.
   2. Demonstrate your method for each pair of values in Exercises 3 − 7 . negative 7 .

Exercises

10. Explain how to write a quadratic equation that has − 6 negative 6  as its only root.
11. Describe the family of quadratic functions that have zeros at r and s. Sketch several members of the family in the coordinate plane.

Find the sum and product of the roots for each quadratic equation.

12. 2 x 2 + 3 x − 2 = 0 2 x squared , plus 3 x minus 2 equals 0
13. x 2 − 2 x + 1 = 0 x squared , minus 2 x plus 1 equals 0
14. x 2 − 5 x + 6 = 0 x squared , minus 5 x plus 6 equals 0

Given the sum and product of the roots, write a quadratic equation in standard form.

15. sum = − 3 ,  product  = − 18 sum equals negative 3 comma . product . equals negative 18
16. sum = 4, product = 3
17. sum = 2, product = 3 4 product , equals , 3 fourths

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