Completing the Square (Part 3)

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

Completing the Square (Part 3)

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Perfect Squares in Two Forms

Elena says, "(x+3)2(x + 3)^{2} can be expanded into x2+6x+9x^{2} + 6x + 9. Likewise, (2x+3)2(2x + 3)^{2} can be expanded into 4x2+6x+94x^{2} + 6x + 9."

1) Find an error in Elena’s statement and correct the error.

2) Show your reasoning.

Perfect in A Different Way

Write each expression in standard form:

3) (4x+1)2(4x + 1)^{2}

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4) (5x2)2(5x - 2)^{2}

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5) (12x+7)2(\frac{1}{2}x + 7)^{2}

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6) (3x+n)2(3x + n)^{2}

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7) (kx+m)2(kx + m)^{2}

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Decide if each expression is a perfect square. If so, write an equivalent expression of the form (kx+m)2(kx + m)^{2}. If not, suggest one change to turn it into a perfect square.

8) Is 4x2+12x+94x^{2} + 12x + 9 a perfect square?

True or false? Write below.

9) If so, write an equivalent expression of the form (kx+m)2(kx + m)^{2}. If not, suggest one change to turn it into a perfect square.

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10) Is 4x2+8x+254x^{2} + 8x + 25 a perfect square?

True or false? Write below.

11) If so, write an equivalent expression of the form (kx+m)2(kx + m)^{2}. If not, suggest one change to turn it into a perfect square.

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When All the Stars Align

Find the value of cc to make each expression a perfect square in standard form. Then, write an equivalent expression in the form of squared factors.

12) Find the value of cc to make 100x2 + 80x + c100x^2 \ + \ 80x \ + \ c a perfect square.

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13) Write an equivalent expression in factored form (kx + m)2(kx \ + \ m)^2

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14) Find the value of cc to make 36x2  60x + c36x^2 \ - \ 60x \ + \ c a perfect square.

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15) Write an equivalent expression in factored form (kx + m)2(kx \ + \ m)^2

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16) Find the value of cc to make 25x2 + 40x + c25x^2 \ + \ 40x \ + \ c a perfect square.

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17) Write an equivalent expression in factored form (kx + m)2(kx \ + \ m)^2

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18) Find the value of cc to make 0.25x2  14x +\c0.25x^2 \ - \ 14x \ + \c a perfect square.

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19) Write an equivalent expression in factored form (kx + m)2(kx \ + \ m)^2

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20) Write your own pair of equivalent expressions.

Solve each equation by completing the square:

21) 25x2+40x=1225x^{2} + 40x = -12

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22) 36x260x+10=636x^{2} - 60x + 10 = -6

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Putting Stars into Alignment

Here are three methods for solving 3x2+8x+5=03x^{2} + 8x + 5 = 0.

Try to make sense of each method.

Method 1:

3x2+8x+5=0(3x+5)(x+1)=0x=53orx=1\begin{array}{rcl} \\[-1em] 3x^2 + 8x + 5 &=& 0 \\[-1em] \\ \\[-1em] (3x + 5)(x + 1) &=& 0 \\[-1em] \\ \end{array} \\ \begin{array}{c} \\[-1em] x = -\frac{5}{3} \quad \text{or} \quad x = -1 \end{array}

Method 2:

3x2+8x+5=09x2+24x+15=0(3x)2+8(3x)+15=0U2+8U+15=0(U+5)(U+3)=0U=5orU=33x=5or3x=3x=53orx=1\begin{array}{rcl} \\[-1em] 3x^2 + 8x + 5 &=& 0 \\[-1em] \\ \\[-1em] 9x^2 + 24x + 15 &=& 0 \\[-1em] \\ \\[-1em] (3x)^2 + 8(3x) + 15 &=& 0 \\[-1em] \\ \\[-1em] U^2 + 8U + 15 &=& 0 \\[-1em] \\ \\[-1em] (U + 5)(U + 3) &=& 0 \\[-1em] \\ \end{array} \\ \begin{array}{rcl} \\[-1em] U = - 5 &\text{or}& U = -3 \\[-1em] \\ \\[-1em] 3x = -5 &\text{or}& 3x = -3 \\[-1em] \\ \\[-1em] x = -\frac{5}{3} &\text{or}& x = -1 \\[-1em] \\ \end{array}

Method 3:

3x2+8x+5=09x2+24x+15=09x2+24x+16=1(3x+4)2=13x+4=1or3x+4=1x=1orx=53\begin{array}{rcl} \\[-1em] 3x^2 + 8x + 5 &=& 0 \\[-1em] \\ \\[-1em] 9x^2 + 24x + 15 &=& 0 \\[-1em] \\ \\[-1em] 9x^2 + 24x + 16 &=& 1 \\[-1em] \\ \\[-1em] (3x + 4)^2 &=& 1 \\[-1em] \\ \end{array} \\ \begin{array}{rcl} \\[-1em] 3x + 4 = 1 &\text{or}& 3x + 4 = -1 \\[-1em] \\ \\[-1em] x = -1 &\text{or}& x = -\frac{5}{3} \\[-1em] \end{array}

Once you understand the methods, use each method at least one time to solve these equations.

23) 5x2+17x+6=05x^{2} + 17x + 6 = 0

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24) 6x2+19x=106x^{2} + 19x = -10

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25) 8x233x+4=08x^{2} - 33x + 4 = 0

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26) 8x226x=218x^{2} - 26x = -21

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27) 10x2+37x=3610x^{2} + 37x = 36

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28) 12x2+20x77=012x^{2} + 20x - 77 = 0

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