Warm Up
Simplifying Radicals
and Solving Quadratic Equations
First, about radicals -
Now we need to learn how tosimplify radicals.
To simplify a radical means to rewrite it as another,
better expression with the same value.
What makes the new version better?
A radical with index 2 (that is, a square root) is simplified
if and only if:
* The radicand does not have a perfect square factor
* The radicand is not a fraction
* There is no radical in the denominator
Simplifying Radicals
When working to simplify radicals you should try to remember some
basic information about perfect squares.
You want to remember:
Perfect Squares
4
9
16
25
36
49
64
81
100
=2x2
=3x3
=4x4
=5x5
=6x6
=7x7
=8x8
=9x9
= 10 x 10
Radicals (square roots)
=2
=3
=4
=5
=6
=7
=8
=9
= 10
We're also going to use these properties to simplify radicals:
Example: simplify the radical
Step 1:
Find the largest perfect square factor of the radicand.
4 is a perfect square, and it's a factor of 48.
But 16 is another perfect square factor, and it's bigger.
16 is the largest perfect square factor of 48.
Step 2: Write the radicand in factored form, using that largest
perfect square factor that you found.
Step 3: Separate each of the factors into its own radical sign.
Step 4: Reduce the "perfect" radical.
The other radical is stuck it can't be simplified any more. But at least it's smaller!
FINAL
ANSWER:
If the number under your radical cannot be divided evenly by any of the perfect
squares, your radical is already in simplest form and cannot be reduced further.
Let's simplify:
2
Your turn . . . .
Sometimes you can use multiplication and division along with
the properties to simplify multiple radicals at once.
Multiply the numbers under the radical to
produce a new radicand...then simplify!
Should you try to simplify or divide first?
Try these problems:
We can use rules of square roots to solve some quadratic equations.
Remember, quadratic equations have two solutions!
But this method ONLY works for equations with no "b" term!
There's no "b" term, so rearrange it as you would a linear equation...
Then, at the last step...simplify the radical and reveal your answer
as "plus or minus"