Discriminant and the Nature of Roots.

Hello again. MERIAN here to fill you in on what we did today in class.

We first started with Mr. K giving us three equations.

f(x) = x² + 5x + 6
y= x² -4x - 5
f(x) = x² + 2x - 15

We were to find:

- The ROOTS
- The sum of the roots
- The ROOT PRODUCT

Here were the answers.

f(x) = x² + 5x + 6

ROOTS: x = -3, x = -2
ROOT SUM: (-2) + (-3) = -5
ROOT PRODUCT: (-2)(-3) = 6

y= x² -4x - 5

ROOTS: x = 5, x = -1
ROOT SUM: 5 + (-1) = 4
ROOT PRODUCT: 5(-1) = -5

f(x) = x² + 2x - 15

ROOTS: x = -5, x = 3
ROOT SUM: (-5) + 3 = -2
ROOT PRODUCT: (-5) 3 = -15

Afterwards, we were to compare our answers to the equation. Remember, mathematics is the science of Pattern--so look for a pattern.

We found out that the pattern here was that the root sum is the negative of B. Let's look at the first one as an example.

f(x) = x² + 5x + 6

ROOT SUM: (-2) + (-3) = -5

AND the other pattern was that The ROOT PRODUCT is = to C.

f(x) = x² + 5x + 6

ROOT PRODUCT: (-2)(-3) = 6

Knowing that there's a relation between them, we can easily figure out the roots of an equation just by simply looking at it. There's a catch though..

Mr. K then gave us another equation and asked us to find the roots of it, the sum of the roots and the product of the roots.

f(x) = x² + 6x + 10

This equation can't be factored out perfectly. it doesn't have REAL roots so it must have an imaginary root. But, since it's not factorable, the other thing we can do is use the Quadratic Formula.

REMINDER: i² = -1 OR i = √-1



x₁ = -3x + i
x2 = -3x - i

^those are your roots. To find out the product of the roots you must distribute -3x - i & -3x + i.
Like so.

( -3x + i )( -3 - i )
9 + 3i - 3i - i²
9 - i²
9 - (-1) = 10 <- The product of roots.

It's different when there is a number in front of a.

Example:

y(x) = 2x² + 5x - 3 <- You can't just take the root sum or root product just by looking at the equation at this point. Since there's a number in front of a it'll factor the other numbers resulting in a different number..

So to do this correctly you must find the coefficients and rewrite 5x. In this case it's going to be 2 & 3.



ROOT SUM
-1 + (-3)/2

-2/2 + (-3)/2

= -5/2

ROOT PRODUCT
(-1)(-3/2)
3/2

We should be also aware of number relationships. Here's a diagram for refrence.


In a quadratic formula, if the inside of a radical is negative then there are no roots. it discriminates the qualities of the equation.

DISCRIMINANTS:
b² - 4ac > 0 = ROOTS
b² - 4ac = 0 = 1 ROOT
b² - 4ac < 0 =" NO">

HOMEWORK IS EXERCISE 16
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