NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 11-13 ALGEBRA I
Lesson 11: Completing the Square
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Lesson 11-13: Completing the Square
Classwork
Opening Exercise
Rewrite the following perfect square quadratic expressions in standard form. Describe patterns in the coefficients for the factored
form, (𝑥 + 𝐴)2, and the standard form, 𝑥2 + 𝑏𝑥 + 𝑐.
FACTORED FORM WRITE THE FACTORS DISTRIBUTE STANDARD FORM
Example: (𝑥 + 1)2
(𝑥 + 2)2
(𝑥 + 3)2
(𝑥 + 4)2
(𝑥 + 5)2
(𝑥 + 20)2
Example
Now try working backward. Rewrite the following standard form quadratic expressions as perfect squares.
STANDARD FORM FACTORED FORM
𝑥2 + 12𝑥+ 36
𝑥2 − 12𝑥+ 36
𝑥2 + 20𝑥+ 100
𝑥2 + 100𝑥+ 2500
𝑥2 − 3𝑥+ 94
𝑥2 + 8𝑥+ 3
Not factorable
NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 11-13 ALGEBRA I
Lesson 11: Completing the Square
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This file derived from ALG I-M4-TE-1.3.0-09.2015
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Exploratory Challenge
Find an expression equivalent to 𝑥2 + 8𝑥+ 3 that includes a perfect square binomial.
Exercises
Rewrite each expression by completing the square.
1. 𝑎2 − 4𝑎 + 15
2. 𝑛2 − 2𝑛 − 15
3. 𝑐2 + 20𝑐 − 40
Process/Tips:
- Add
- Don’t change the value of the
expression.
NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 11-13 ALGEBRA I
Lesson 11: Completing the Square
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4. 𝑥2 − 1000𝑥 + 60 000
5. 𝑘2 + 7𝑘 + 6
6. 𝑧2 − 0.2𝑧 + 1.5
7. 𝑥2 − 𝑏𝑥 + 𝑐
Lesson Summary
Just as factoring a quadratic expression can be useful for solving a quadratic equation, completing the square also
provides a form that facilitates solving a quadratic equation.
NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 11-13 ALGEBRA I
Lesson 11: Completing the Square
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Example Now complete the square for 2𝑥2 + 16𝑥 + 3.
Exercises
For Exercises 1–3, rewrite each expression by completing the square.
1. 3𝑥2 + 12𝑥 − 8
2. 4𝑝2 − 12𝑝 + 13
3𝑥2 − 18𝑥 − 2
3(𝑥2 − 6𝑥) − 2
3(𝑥2 − 6𝑥 + 9) − 3(9) − 2
3(𝑥 − 3)2 − 3(9) − 2
3(𝑥 − 3)2 − 29
Lesson Summary
Here is an example of completing the square of a quadratic expression of the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐.
NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 11-13 ALGEBRA I
Lesson 11: Completing the Square
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This file derived from ALG I-M4-TE-1.3.0-09.2015
This work is licensed under a
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Classwork Solve the equation for 𝑏: 2𝑏2 − 9𝑏 = 3𝑏2 − 4𝑏 − 14.
Example 1
Solve for 𝑥.
12 = 𝑥2 + 6𝑥
Rational and Irrational Numbers
The sum or product of two rational numbers is always a rational number.
The sum of a rational number and an irrational number is always an irrational number.
The product of a rational number and an irrational number is an irrational number as long as the rational number is not zero.
Example 2
Solve for 𝑥.
4𝑥2 − 40𝑥 + 94 = 0
NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 11-13 ALGEBRA I
Lesson 11: Completing the Square
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Exercises
Solve each equation by completing the square.
8. 𝑥2 − 2𝑥 = 12
9. 12 𝑟
2 − 6𝑟 = 2
10. 2𝑝2 + 8𝑝 = 7
Lesson Summary When a quadratic equation is not conducive to factoring, we can solve by completing the square.
Completing the square can be used to find solutions that are irrational, something very difficult to do by factoring.
