Download Lesson 3.9 Solving quadratics using Square roots

Lesson 3.9 Solving quadratics using Square roots
Learning Objectives: SWBAT
1.
Simplify Square Root (radical) expressions by using perfect square factors and by
rationalizing the denominator
2.
Solve quadratic equations using the properties of square roots
Making a connection:
• Because quadratics are always "squared" we must remember some things about
"square roots" when it comes to solving:
• The above a fancy way of saying that all numbers have two square roots: A
positive and a negative
> Example: √4 = 2 and -2 (also can be written as ±2)
Simplifying Square roots
• When solving quadratics using square roots, we also need to remember how how to
simplify square roots using radical form (from Algebra 1 or Geometry). NO
DECIMALS
> Radical form is more accurate/exact than decimals
> A square root is in its simplest radical form if the number under the square
root sign does not have any perfect squares as factors
> Therefore the first step in simplifying radical numbers is to factor the number
under the square root sign using a perfect square
Example: Simplify √8
– Use a perfect square to factor the number under the radical sign
– Any perfect square under the square root sign equals its square root (radical
sign goes away)
Simplifying Radicals Practice
Lesson 3.9 Solving quadratics using Square roots
Rationalizing the denominator
• Many of the solutions that we will be dealing with in the remainder of this unit will be
fractions (sorry).
• When dealing with fractions and square roots in the same expression, the
expression is not simplified if there is a radical number in the denominator.
• If we ever see an expression that has a radical number in the denominator, we get
rid of it through a process called "Rationalizing the Denominator"
Example:
Multiply by
"fancy form of
the number "1"
Rationalizing the denominator practice
Radical sign
goes away
Lesson 3.9 Solving quadratics using Square roots
When do you solve quadratic equations using square roots?
• When b = 0 (in other words, there is no x1 term)
What is the process for solving quadratics using square roots?
1.
Isolate the x2 term
2.
Take the square root of both sides of the equation
3.
Simplify you answer (as we practiced on the previous two pages)
How do you know you are finished with the process?
• The answer is in simplest radical form
• There are no radicals in the denominator
• Remember, your answer will have a positive and negative square root (±)
Example 1: Solve 4x2 + 13 = 253
• Objective: Isolate the x2 term and take the square root of both sides
> Subtract 13 from both sides: 4x2 = 240
> Divide both sides by 4: x2 = 60 (the x2 term is now isolated)
> Take square rood of both sides: x = ±√60
> Simplify: x = ±2√15
Example 2: Solve 3x2 + 10 = 20
• Objective: Isolate the x2 term and take the square root of both sides
> Subtract 10 from both sides: 3x2 = 10
> Divide both sides by 3: x2 = 10/3
> Take square rood of both sides: x = ±√10/√3
> Simplify by rationalizing denominator: x = ±√30/3
Your Turn #1: Solve the following quadratic equations by using square roots.
remember to simplify you answer
Your Turn #2: Solve (hint, make "a" positive)
Lesson 3.9 Solving quadratics using Square roots
Practice: Solve the following quadratic equations, leave your answer in simplest
radical form
Lesson 3.9 Solving quadratics using Square roots
Practice: Solve the following quadratic equations, leave your answer in simplest
radical form