Completing the Square (Part 1)

ID: dalaz-zizaj
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
Standards: HSA-SSE.AHSA-SSE.A.2HSA-REI.B.4.aHSA-REI.B.4.b
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18 questions

Completing the Square (Part 1)

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Perfect or Imperfect?

1) Select all\textbf{all} expressions that are perfect squares. Write each corresponding letter in the answer box and separate letters with commas.

a) (x+5)(5+x)(x + 5)(5 + x) \quad \quad\quad b) (x+5)(x5)(x + 5)(x - 5) \quad \quad c) (x3)2(x - 3)^{2} \quad \quad d) x32x - 3^{2} \quad \quad e) x2+8x+16x^{2} + 8x + 16 \quad \quad\quad f) x2+10x+20x^{2} + 10x + 20 \quad \quad

2) Explain how you know.

Building Perfect Squares

Study the table. Then write the complete standard form or factored form.

Form IDStandard FormFactored FormAx2+6x+9Bx210x+25C(x7)2Dx220x+(x)2Ex2+16x+(x+)2Fx2+7x+(x+)2Gx2+bx+(x+)2\begin{array}{|c|c|c|} \hline \\[-1em] \textbf{Form ID} & \textbf{Standard Form} & \textbf{Factored Form} \\[-1em] \\ \hline \\[-1em] \textbf{A} & x^{2} + 6x + 9 & \\[-1em] \\ \hline \\[-1em] \textbf{B} & x^{2} - 10x + 25 & \\[-1em] \\ \hline \\[-1em] \textbf{C} & & (x - 7)^{2} \\[-1em] \\ \hline \\[-1em] \textbf{D} & x^{2} - 20x + \underline{\quad} & (x - \underline{\quad})^{2} \\[-1em] \\ \hline \\[-1em] \textbf{E} & x^{2} + 16x + \underline{\quad} & (x + \underline{\quad} )^{2} \\[-1em] \\ \hline \\[-1em] \textbf{F} & x^{2} + 7x + \underline{\quad} & (x + \underline{\quad} )^{2} \\[-1em] \\ \hline \\[-1em] \textbf{G} & x^{2} + bx + \underline{\quad} & (x + \underline{\quad} )^{2} \\[-1em] \\ \hline \end{array} 

3) Factored Form A

4) Factored Form B

5) Standard Form C

6) Standard Form D

7) Factored Form D

8) Standard Form E

9) Factored Form E

10) Standard Form F

11) Factored Form F

12) Standard Form G

13) Factored Form G

Dipping Our Toes in Completing the Square

One technique for solving quadratic equations is called completing the square\textbf{completing the square}. Here are two examples of how Diego and Mai completed the square to solve the same equation.

Diego:Mai:x2+10x+9=0x2+10x=9x2+10x+25=9+25x2+10x+25=16(x+5)2=16x+5=4orx+5=4x=1orx=9x2+10x+9=0x2+10x+9+16=16x2+10x+25=16(x+5)2=16x+5=4orx+5=4x=1orx=9\begin{array}{cc} \text{Diego:} & \text{Mai:} \\ \begin{array}{rcl} \\[-1em] x^2 + 10x + 9 &=& 0 \\[-1em] \\ \\[-1em] x^2 + 10x &=& -9 \\[-1em] \\ \\[-1em] x^2 + 10x + 25 &=& -9 + 25 \\[-1em] \\ \\[-1em] x^2 + 10x + 25 &=& 16 \\[-1em] \\ \\[-1em] (x + 5)^2 &=& 16 \\[-1em] \\ \\[-1em] x + 5 = 4 &\text{or}& x + 5 = -4 \\[-1em] \\ \\[-1em] x = -1 &\text{or}& x = -9 \\[-1em] \end{array} & \begin{array}{rcl} \\[-1em] x^2 + 10x + 9 &=& 0 \\[-1em] \\ \\[-1em] x^2 + 10x + 9 + 16 &=& 16 \\[-1em] \\ \\[-1em] x^2 + 10x + 25 &=& 16 \\[-1em] \\ \\[-1em] (x + 5)^2 &=& 16 \\[-1em] \\ \\[-1em] x + 5 = 4 &\text{or}& x + 5 = -4 \\[-1em] \\ \\[-1em] x = -1 &\text{or}& x = -9 \\[-1em] \end{array}\end{array}

Study the worked examples. Then, try solving these equations by completing the square:

14) x2+6x+8=0x^{2} + 6x + 8 = 0

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15) x2+12x=13x^{2} + 12x = 13

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16) 0=x210x+210 = x^{2} - 10x + 21

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17) x22x+3=83x^{2} - 2x + 3 = 83

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18) x2+40=14xx^{2} + 40 = 14x

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