Your Federal Quarterly Tax Payments are due April 15th

# Creating Quadratic Equations By Using Their Roots by pyb17727

VIEWS: 37 PAGES: 2

• pg 1
```									Creating Quadratic Equations By Using Their Roots

• In our previous work, we’ve been given a quadratic equation
functions, we can also create a quadratic equation by using
only its roots or zeros. If x = r is a root, then (x − r) is a factor of
the equation. Therefore, if an equation has roots x = r1 and
x = r2 , then the resulting equation is (x − r1 )(x − r2 ) = 0 .
• Example:
Find a quadratic equation with the following roots:
1
(a.) − and 3 (b.) 2 − 3 and 2 + 3 (c.) 1 − 2i and 1 + 2i
2

1
(a.) Since the roots are x = −     and x = 3 , then the factors are
2
⎛    1⎞
⎜ x + ⎟ and (x − 3) . Therefore, we have the equation:
⎝    2⎠
⎛     1⎞
⎜ x + 2 ⎟ (x − 3) = 0
⎝       ⎠
(2x + 1)(x − 3) = 0    ← Multiply by 2
(b.) Since the roots are x = 2 − 3 and x = 2 + 3 , then the
factors are (x − 2 + 3 ) and (x − 2 − 3 ) . Therefore, we have
the equation:
(x − 2 + 3 )(x − 2 − 3 ) = 0
x 2 − 2x − x 3 − 2x + 4 + 2 3 + x 3 − 2 3 − 3 = 0
x 2 − 2x − x 3 − 2x + 4 + 2 3 + x 3 − 2 3 − 3 = 0
x 2 − 2x − 2x + 4 − 3 = 0
x 2 − 4x + 1 = 0
(c.) Since the roots are x = 1 − 2i and x = 1 + 2i , then the factors
are (x − 1 + 2i) and (x − 1 − 2i) . Therefore, we have the equation:
(x − 1 + 2i)(x − 1 − 2i) = 0
x 2 − x − 2ix − x + 1 + 2i + 2ix − 2i − 4i2 = 0
x 2 − x − 2ix − x + 1 + 2i + 2ix − 2i − 4( −1) = 0    ← i2 = −1
x 2 − 2x + 5 = 0
The Sum and Product of Roots

• In the previous section, we discovered how to work backwards
to determine an equation using the given roots. We can also
create an equation from the sum and product of its roots.
• Any quadratic equation ax 2 + bx + c = 0 can be expressed in
the form

x 2 − (sum of the roots)x + (product of the roots) = 0

where:

b
sum of the roots = −
a
c
product of the roots =
a

• Example:
Calculate the sum and product to determine a quadratic
equation with roots x = 2 ± 3 .

Solution:

The roots are x = 2 − 3 and x = 2 + 3 .

Therefore, the sum of the roots is:
(2 − 3 ) + (2 + 3 ) = 4

The product of the roots is:
(2 − 3 )(2 + 3 )
= 4+2 3 −2 3 −3
=1

Therefore, the quadratic equation is x 2 − 4x + 1 = 0 .

```
To top