No more than two
A quadratic equation has at most two roots.
How many numbers work?
The equation x² − 5x + 6 = 0 has x = 2 as a root. Is that the only number? Could there be many?
How many different numbers can make a quadratic equation true?
What this lesson covers
The idea
Since a quadratic polynomial has at most two zeroes, a quadratic equation has at most two roots.
How many numbers work?
The equation x² − 5x + 6 = 0 has x = 2 as a root. Is that the only number? Could there be many?
How many different numbers can make a quadratic equation true?
- Only one
- At most two
- As many as we like
Test the whole row
Test every whole number from −3 to 7 in each equation, and count how many make the left side 0. The equation is also shown as a product of factors: a product is 0 only when one of its factors is 0.
Never more than two
A quadratic polynomial has at most two zeroes. The roots of a quadratic equation are the zeroes of its polynomial. So a quadratic equation has at most two roots: two, one, or none.
Two roots: x² − 5x + 6 = 0 has 2 and 3. One root: x² − 6x + 9 = 0 has only 3. None: x² + 4 = 0 has no real root, because x² + 4 is never 0.
When (x − 2)(x − 3) = 0, the product is 0 only if x − 2 = 0 or x − 3 = 0. Two factors give at most two roots.
Notes
A quadratic polynomial has at most two zeroes, so a quadratic equation has at most two roots: two, one, or none.
Check yourself
(x − 4)(x + 1) = 0. What is the positive root?
Answer: 4
x − 4 = 0 gives x = 4, and x + 1 = 0 gives x = −1. The equation has just these two roots.
How many different numbers are roots of x² − 6x + 9 = 0? (Hint: it is (x − 3)(x − 3) = 0.)
Answer: 1
Both factors give x = 3. So there is one root, 3, counted twice.
A friend says x² − 7x + 12 = 0 has three roots: 3, 4 and 5. Why must this be wrong?
How many real roots does x² + 4 = 0 have?
Answer: 0
x² is 0 or more, so x² + 4 is at least 4. It is never 0, so there is no real root. That is still "at most two".
- A quadratic equation has at most two roots — correct. Yes! It has two: 3 and 4. And 5 is not a root: 25 − 35 + 12 = 2.
- Roots must be even numbers. There is no such rule. The roots 3 and 4 are fine.
- A quadratic equation has exactly one root. It can have two: 3 and 4 both work (9 − 21 + 12 = 0 and 16 − 28 + 12 = 0).