- (x 2/3)^2 4/9 1/3 0
- (x 2/3)^2 1/9 0 (add together the 4/9 1/3 by converting the denomina-
tor to 9, in which 1/3 becomes 3/9) - (x 2/3)^2 1/9 (move the 1/9 to the other side of the equation by subtracting
it from both sides) - x 2/3 1/3 or x 2/3 1/3
That means that x 1 or x 1/3 are the two roots that make the equation true
(just substitute either number into the initial equation to see that they are both true).
What are examples of polynomialswith one root and no roots?
The following is an example of a polynomial with only one root:
x^2 6x 9 0
(x^2 6x (6/2)^2 ) (6/2)^2 9 0
(x 3)^2 9 9 0
(x 3)^2 0
x 3 0
x 3, or the polynomial has only one root x 3
But not all polynomials have roots. The following is an example of a polynomial
with no root:
2x^2 6x 8 0
(1/2)(2x^2 6x 8) (1/2)0
x^2 3x 4 0
(x^2 3x (3/2)^2 ) (3/2)^2 4 0
(x 3/2)^2 9/4 4 0
(x 3/2)^2 7/4 0
(x 3/2)^2 7/4
Because a real number squared is greater than or equal to 0, that means (x 3/2)^2
will always be greater than or equal to 0. Thus, the answer can’t be 7/4, a negative
number, and there are no real roots for this polynomial.What is a quadratic equation?
A quadratic equation is a second-degree (order) polynomial equation, thus it is guar-
anteed to have two solutions, both of which may be real or complex. This is seen in the
standard form ax^2 bxc0.
The roots of xcan be found in the equation by factoring and completing the
154 square (a method of transforming a quadratic equation so that it is in the form of a