Completing the Square (Part 1-3)
Lesson 12-14
Unit 7
Quadratic Equations
HSA-REI.B.4.a: Use the method of completing the square to transform any quadratic equation in 𝘹 into an equation of the form (𝘹 – 𝘱)² = 𝘲 that has the same solutions. Derive the quadratic formula from this form.
HSA-REI.B.4.b: Solve quadratic equations by inspection (e.g., for 𝘹² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation.
HSA-SSE.A: Interpret the structure of expressions.
HSA-SSE.A.2: Use the structure of an expression to identify ways to rewrite it. For example, see 𝘹⁴ – 𝘺⁴ as (𝘹²)² – (𝘺²)², thus recognizing it as a difference of squares that can be factored as (𝘹² – 𝘺²)(𝘹² + 𝘺²).
HSA-REI.A: Understand solving equations as a process of reasoning and explain the reasoning.
Expressions and Equations
Perfect or Imperfect?
Unit 7 ● Lesson 12
x² + 10x = -20
x² + 10x + 25 = 5
(x + 5)² = 5
(x + 5)² - 5 = 0
Warm-up
Page 254
Page 240
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
Unit 7 ● Lesson 12
Let’s learn a new method for solving quadratic equations.
We can explain what it means to “complete the square” and describe how to do it so that we can solve quadratic equations by completing the square.
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
Where were we? Where are we? Where are we going?
Unit 6 ● Lesson 12-13
Agenda Review
You are successful today when...,
● You can explain what it means to “complete the square” and describe how to do it.
● You can solve quadratic equations by completing the square and finding square roots.
● When given a quadratic equation in which the coefficient of the squared term is 1, you can solve it by completing the square.
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
Both sides in an equation
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
Remember you can divide all terms by “a” to make “a” = 1.
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
12.2 Activity: Building Perfect Squares
I can solve quadratic equations by completing the square
10 mins Total
3 mins individual - 2 mins group - 3 mins class share
pg 240
(x + 3)²
(x - 5)²
x² -14x + 49
100
10
64
8
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
Glossary
I can solve quadratic equations by completing the square
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
12.3 Activity: Dipping Our Toes in Completing the Square
I can solve quadratic equations by completing the square
How are the two solution methods alike? How they are different?
Either method works, but some people prefer Diego’s approach because moving the original constant term to the other side of the equal sign allows them to see what constant term is needed to make a perfect square on the left side. They also find it to be less prone to errors.
(x + 5)² -16 = 0
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
12.3 Activity: Dipping Our Toes in Completing the Square
I can solve quadratic equations by completing the square
15 mins Total
4 mins individual - 4 mins group - 4 mins class share
pg 241-2
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
12.3 Activity: Dipping Our Toes in Completing the Square
I can solve quadratic equations by completing the square
15 mins Total
4 mins individual - 4 mins group - 4 mins class share
pg 241-2
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
12.3 Activity: Dipping Our Toes in Completing the Square
I can solve quadratic equations by completing the square
15 mins Total
4 mins individual - 4 mins group - 4 mins class share
pg 241-2
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
Unit 7 ● Lesson 13
Let’s solve some harder quadratic equations.
We can explain what it means to “complete the square” and describe how to do it so that we can solve quadratic equations by completing the square.
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
Perfect or Imperfect?
Unit 7 ● Lesson 13
Warm-up
Page 247
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
13.2 Activity: Solving Some Harder Equations
I can solve quadratic equations by completing the square
15 mins Total
5 mins individual - 4 mins group - 5 mins class share
pg 247-8
1 - 4 only
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
13.2 Activity: Solving Some Harder Equations
I can solve quadratic equations by completing the square
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
Completing the Square (Part 1)
I can solve quadratic equations by completing the square
Yes. (x + 9)(x + 1)
NO
NO
Yes
Not all equations can be written in factored form so it is not always possible to solve that way. In those cases, we can solve by completing the square
(when a = 1).
Lesson Synthesis
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
Make It a Perfect Square
I can solve quadratic equations by completing the square
8
Cool-down
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
We will solve quadratics by completing the square in order to transform the equation to the form (x-h)² = k so that we can further develop our critical thinking and quantitative analysis skills for the real world.
Given the function y = x² +12x + 32
32
x = -8, -4
(x+6)² -4
(-6, -4)
You could also use -b/2a (-6) to find the aos; then input -6 into the function to find the y-value of the vertex) 36-72+32 = -4
Unit 7 ● Lesson 14
Let’s complete the square for some more complicated expressions.
We can explain what it means to “complete the square” and describe how to do it so that we can solve quadratic equations by completing the square.
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
14.2 Activity: Perfect in A Different Way
I can solve quadratic equations by completing the square
15 mins Total
5 mins individual - 5 mins group - 5 mins class share
pg 255
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
14.2 Activity: Perfect in A Different Way
I can solve quadratic equations by completing the square
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
14.3 Activity: When All the Stars Align
I can solve quadratic equations by completing the square
a is k²
k is √a
c is m²
m is √c
b is 2km
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
14.3 Activity: When All the Stars Align
I can solve quadratic equations by completing the square
15 mins Total
5 mins individual - 5 mins group - 5 mins class share
pg 256
C = [80/(2√100)]²
C = [60/(2√36)]²
C = [40/(2√25)]²
C = [14/(2√¼)]²
C = [b/(2√a)]²
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
14.3 Activity: When All the Stars Align
I can solve quadratic equations by completing the square
25x² +40x + 16 = 4
(5x +4)² = 4
5x + 4 = ±2
5x = -4 + 2 or 5x = -4 -2
5x = -2 or 5x = -6
36x² -60x = -16
36x² -60x + 25 = 9
(6x - 5)² = 9
6x - 5 = ±3
6x = 5 + 3 or 6x = 5 -3
6x = 8 or 6x = 2
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
14.3 Activity: When All the Stars Align
I can solve quadratic equations by completing the square
C = [b/(2√a)]²
C is some number squared, so it will always be positive.
25 would make it a perfect square, so we can either add 15 to each side of the equation, or first subtract 10 from each side and then add 25.
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
Completing the Square (Part 3)
I can solve quadratic equations by completing the square
What number would you add to make the left side a perfect square?
64 = [32/(2√4)]²
It’s easier when the coefficient of the squared term is 1 or another perfect square. It’s harder when some of the coefficients are fractions or the quadratic term’s coefficient is not a perfect square.
C = [b/(2√a)]²
Lesson Synthesis
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
One More Equation
I can solve quadratic equations by completing the square
49 = [28/(2√4)]²
8
Cool-down
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
Unit 6 ● Lesson 12-14
● I can explain what it means to “complete the square” and describe how to do it.
● I can solve quadratic equations by completing the square and finding square roots.
● When given a quadratic equation in which the coefficient of the squared term is 1, I can solve it by completing the square.
● I can complete the square for quadratic expressions in standard form when a is not 1 and explain the process.
● I can solve quadratic equations in which the squared term coefficient is not 1 by completing the square.
Learning
Targets
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.
Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.