Chapter 9
9.1 Radical Review
9.2 Introduction to Circles
9.3 Altitude on Hypotenuse Theorem
9.4 Pythagorean Theorem
9.5 Distance Formula and Coordinate Geometry
9.6 Pythagorean Triples
9.7 Special Right Triangles
9.8 Space figures and the Pythagorean Theorem
9.9 Introduction to Trigonometry
9.1 Radical Review
You must be able to simplify radicals. A radical consists of three parts.
1) The radical sign, 2) the index, and 3) the radicand.
The index is 3,
the radicand is 8
and the radical sign is √
The example above means that you are looking for a single factor that when multiplied times itself three times
will give you an answer of 8. In this case that factor is 2.
NOTE: An index of 2 is implied by the √ symbol and the index number is not shown in this case.
The radical without a visible index is asking for the square root.
Example:
This is asking ? • ? = 25.
The square root of 25 is 5, since 5 • 5 =
When the index on the radical is 3
Example:
= 25!
, you are expected to find the cubed root.
This is asking ? • ? • ? = 8.
This expression means that the factor you are asked to find, when multiplied by itself three times, results in the
radicand.
The cubed root of 8 is 2, since 2 • 2 • 2 =
!
In summary, you will be taking roots of numbers which means you are looking for a factor when multiplied
with itself the number of times that the index indicates, will result in the radicand value.
Simplest Form: There will often be times that you will NOT find a whole number that will give you the exact
amount of your radicand. In these cases you must put your answer in simplest form. Below are rules for
simplifying radicals with examples of each.
Rules for Simplifying Radicals by Counterexample:
Rule 1: A radical expression is not in simplest form if any factors of the radicand are perfect squares.
Not simplest form :
7•5•
Example:
35
Simplest Form!
By divisibility rules, we know that 3 is a factor of 3675.
(The sum of the digits is 21)
3675 ÷ 3 = 1225.
1225 = 49 • 25! Two perfect squares!!! Yea!
So 3675 can be factored into 3 • 49 • 25
Rule 2: A radical expression is not in simplest form if any fractions are left as radicands.
Examples:
A) Not simplest form:
B) Not simplest form:
simplest form!
simplest form!
Step 1) Apply radical to each part of fraction
Step 2) If you can take the root of the parts, do it! If your new fraction can be further simplified, then write in simplest form.
Step 3) If the radicands of the numerator and denominator are not perfect squares, simplify them as much as possible!
Rule 3: A radical expression is not in simplest form if the denominator contains a radical.
The process of eliminating a radical in the denominator is referred to as “rationalizing the denominator.” In
other words, you must make sure the denominator contains a rational number, not a radical value.
You should remember doing all of this in Algebra last year!
Example:
1
2
3
4
5
6
NOTE: The example above includes ALL of the steps involved for writing a radical expression in simplest form!
1.
2.
3.
4.
5.
6.
Apply radical to numerator and denominator separately.
Rationalize the denominator
Multiply
Look for perfect square factors under radical in numerator, and factor it.
Take square root of any perfect square factors under radical and write as coefficient of radical.
Look for common rational factors of numerator and denominator, and simplify completely!
Rules for Operations on Radicals
Rule 1: To add or subtract radicals.  The radicands must be the same.
 Keep the radicand and add or subtract the coefficients.
This process is similar to the way you add or subtract monomials that are alike:
Remember to group like terms: 2x + 25y + 4x + 8y + 5 = (2x + 4x) + (25y + 8y) + 5 = 6x + 33y + 5
Example 1:
=
(Don’t forget these guys have a coefficient of 1)
Sometimes you must simplify the radicals before adding or subtracting. See next example.
Example 2:
=
(3 • 9) (2 • 25)
(3 • 36)
Rule 2: To multiply or divide radicals:
 Multiply or divide the coefficients  Multiply or divide radicands  Simplify if possible.
Multiplication Examples:
A) Find the product:
=
=
=
=
=
=
simplest form!
B) Find the product:
(Put both under same radical)
=
(Simplify as much as possible)
=
(No fractions as radicands!)
=
=
simplest form!
Radical Division
Remember that no matter what they contain, if the numerator and denominator are the same…
… (all) = 1!
Examples:
A) Find the quotient:
18 = 2 • 9
75 = 3 • 25
=
=
=
=
simplest form!
B) Find the quotient:
=
Denominator alert!
=
=
=2
simplify!
simplest form!
Quadratic Equations
Guidelines for FACTORING Quadratic Equations COMPLETELY
1.
2.
3.
4.
5.
6.
7.
Look for GCF of all terms, and factor it out if you find one.
Look for a difference of squares.
Look for a perfect square trinomial.
If a trinomial is not a square (see steps 2 & 3), look for a pair of binomial factors.
If a polynomial has four or more terms, look for a way to group the terms in pairs.
Make sure each binomial or trinomial factor is prime.
Check your work by multiplying the factors.
Solving Quadratic Equations
Step 1: Set the equation equal to zero.
Step 2: Use factoring or the quadratic formula to solve for “x” value(s).
Step 3: If you used factoring, set each factor equal to zero and solve each for “x.”
Example:
Step 1:
Step 2:
Some of you may have learned to factor by the multiplying up method.
Multiply the coefficient of the quadratic term by the constant term (first • last terms)
(18) (- 20) = - 360
Find factors of -360 that have a sum equal to the coefficient of the middle term, -9.
(-24) (15) = - 360
Next, write two binomials using the coefficient of the first term in both.
(18x - 24) (18x + 15)
Since we used an extra 18, we must divide each binomial by factors of 18 to get rid of the extra.
=
Step 3:
(3x - 4) (6x + 5)
(3x – 4) = 0
and/or
(6x + 5) = 0
3x = 4
and/or
6x = - 5
Check Factors: (3x – 4)(6x + 5) =
(check!)
If there is no linear term (“x” term), you can isolate the squared term on one side of the equation
and take the square root of both sides.
Remember, you are solving an equation, so be sure to give both signs of your result!
Examples:
A)
(isolate the squared term on one side)
(take the square root of each side)
(both values of “x”)
B)
(since the coefficient of the squared term is also a square, leave it)
(take the square root of each side)
(solve for x)
(both values of “x”)
Using the Quadratic Formula:
1. First, your equation must be in the form: Ax2 + By + C = 0
2. Substitute A, B, and C values from your equation into the quadratic formula.
3. Solve for any real roots.
Example:
Equation:
A=2
B=-6
This part . . .
C=-8
. . . plus and minus . . .

. . . the value under the radical!
The discriminant…
(Two real roots)
and
If we had factored out the GCF first as we could have done:
We would have this trinomial 
and using reverse foil to factor,
setting each binomial equal to zero:
Note: Our variable was “y,” not “x” this time!
(y – 4)(y + 1) = 0
y – 4 = 0 and y + 1 = 0
y = {4, -1} 