What are Perfect Squares?: IM Alg1.7.11
[url=https://im.kendallhunt.com/HS/teachers/1/7/11/preparation.html]“What are Perfect Squares?”[/url] from IM Algebra I © 2019 [url=https://www.illustrativemathematics.org/]Illustrative Mathematics[/url]. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0[/url] license.
Table of Contents
- Alg1.7.11 Lesson
- IM Alg1.7.11 Lesson: What are Perfect Squares?
- Alg1.7.11 Practice
- IM Alg1.7.11 Practice: What are Perfect Squares?
IM Alg1.7.11 Lesson: What are Perfect Squares?
[size=150]In each equation, what expression could be substituted for [math]a[/math] so the equation is true for all values of [math]x[/math]?[/size][br][br][math]x^2=a^2[/math]
[math]\left(3x\right)^2=a^2[/math]
[math]a^2=7x\cdot7x[/math]
[math]25x^2=a^2[/math]
[math]a^2=\frac{1}{4}x^2[/math]
[math]a^2=\left(x+1\right)^2[/math]
[math]\left(2x-9\right)\left(2x-9\right)=a^2[/math]
Each expression is written as the product of factors. Write an equivalent expression in standard form.
[math]\left(3x\right)^2[/math]
[math]7x\cdot7x[/math]
[math](x+4)(x+4)[/math]
[math]\left(x+1\right)^2[/math]
[math]\left(x-7\right)^2[/math]
[math]\left(x+n\right)^2[/math]
[size=150]Why do you think the following expressions can be described as [b]perfect squares[/b]?[/size][br][br][table][tr][td][math]x^2+6x+9[/math] [/td][td][math]x^2-16x+64[/math] [/td][td][math]x^2+\frac{1}{3}x+\frac{1}{36}[/math] [/td][/tr][/table]
Write each expression in factored form.
[math]x^4-30x^2+225[/math]
[math]x+14\sqrt{x}+49[/math]
[math]5^{2x}+6\cdot5^x+9[/math]
Han and Jada solved the same equation with different methods. Here they are:
[table][tr][td][size=150]Han’s method:[/size][br][/td][td][size=150]Jada’s method[/size]:[/td][/tr][tr][td][math]\displaystyle \begin {align} (x-6)^2&=25\\(x-6)(x-6)&=25 \\x^2-12x+36&=25\\ x^2-12x+11&=0\\(x-11)(x-1)&=0\\ \\x=11 \quad \text{or} \quad x&=1 \end{align}[/math][br][/td][td][math]\displaystyle \begin {align} (x-6)^2&=25\\ \\x-6=5 \quad &\text{or} \quad x-6=\text-5\\ x=11 \quad &\text{or} \quad x=1 \end{align}[/math][/td][/tr][/table][br][size=150]Work with a partner to solve these equations. For each equation, one partner solves with Han’s method, and the other partner solves with Jada’s method. Make sure both partners get the same solutions to the same equation. If not, work together to find your mistakes.[/size]
IM Alg1.7.11 Practice: What are Perfect Squares?
[size=150]Select [b]all[/b] the expressions that are perfect squares.[/size]
Each diagram represents the square of an expression or a perfect square. Complete the cells in the last table.
How are the contents of the three diagrams alike?
[size=150]This diagram represents [math](\text{term_1 + term_2})^2[/math]. [/size][br][center][img]data:image/png;base64,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[/img][/center][br]Describe your observations about cells 1, 2, 3, and 4.
Rewrite the perfect-square expression [math]\left(n+7\right)^2[/math] in standard form: [math]ax^2+bx+c[/math].[br]
Rewrite the perfect-square expression [math]\left(5-m\right)^2[/math] in standard form: [math]ax^2+bx+c[/math].[br]
Rewrite the perfect-square expression [math]\left(h+\frac{1}{3}\right)^2[/math] in standard form: [math]ax^2+bx+c[/math].[br]
How are the [math]ax^2[/math], [math]bx[/math], and [math]c[/math] of a perfect square in standard form related to the two terms in [math](\text{term_1 + term_2})^2[/math]?[br]
Solve each equation.
[math]\left(x-1\right)^2=4[/math]
[math]\left(x+5\right)^2=89[/math]
[math]\left(x-2\right)^2=0[/math]
[math]\left(x+11\right)^2=121[/math]
[math]\left(x-7\right)^2=\frac{64}{49}[/math]
[size=150]Explain or show why the product of a sum and a difference, such as [math]\left(2x+1\right)\left(2x-1\right)[/math], has no linear term when written in standard form.[/size]
[size=150]To solve the equation [math](x+3)^2=4[/math], Han first expanded the squared expression.[br][br]Here is his incomplete work:[/size][br][math]\begin{align}(x+3)^2&=4\\ (x+3)(x+3)&=4\\ x^2+3x+3x+9&=4\\ x^2+6x+9&=4 \end{align}[/math][br][br]Complete Han’s work and solve the equation.
[size=150]Jada saw the equation [math]\left(x+3\right)^2=4[/math] and thought, “There are two numbers, 2 and -2, that equal 4 when squared. This means [math]x+3[/math] is either 2 or it is -2. I can find the values of [math]x[/math] from there.”[/size][br][br]Use Jada’s reasoning to solve the equation.
Can Jada use her reasoning to solve [math]\left(x+3\right)\left(x-3\right)=5[/math]? Explain your reasoning.[br]