# Completing the Square (Part 3)

ID: hovan-lilik
Illustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Standards: HSA-SSE.A.2HSA-REI.B.4.aHSA-REI.B.4.b

28 questions

# Completing the Square (Part 3)

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##### Perfect Squares in Two Forms

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

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

##### Perfect in A Different Way

Write each expression in standard form:

3) ﻿$(4x + 1)^{2}$﻿

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4) ﻿$(5x - 2)^{2}$﻿

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5) ﻿$(\frac{1}{2}x + 7)^{2}$﻿

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6) ﻿$(3x + n)^{2}$﻿

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7) ﻿$(kx + m)^{2}$﻿

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

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

True or false? Write below.

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

Show Work

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

True or false? Write below.

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

Show Work
##### When All the Stars Align

Find the value of ﻿$c$﻿ 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 ﻿$c$﻿ to make ﻿$100x^2 \ + \ 80x \ + \ c$﻿ a perfect square.

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

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

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

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

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

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

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

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20) Write your own pair of equivalent expressions.

Solve each equation by completing the square:

21) ﻿$25x^{2} + 40x = -12$﻿

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22) ﻿$36x^{2} - 60x + 10 = -6$﻿

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##### Putting Stars into Alignment

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

Try to make sense of each method.

Method 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:

﻿$\begin{array}{rcl} \\[-1em] 3x^2 + 8x + 5 &=& 0 \\[-1em] \\ \\[-1em] 9x^2 + 24x + 15 &=& 0 \\[-1em] \\ \\[-1em] (3x)^2 + 8(3x) + 15 &=& 0 \\[-1em] \\ \\[-1em] U^2 + 8U + 15 &=& 0 \\[-1em] \\ \\[-1em] (U + 5)(U + 3) &=& 0 \\[-1em] \\ \end{array} \\ \begin{array}{rcl} \\[-1em] U = - 5 &\text{or}& U = -3 \\[-1em] \\ \\[-1em] 3x = -5 &\text{or}& 3x = -3 \\[-1em] \\ \\[-1em] x = -\frac{5}{3} &\text{or}& x = -1 \\[-1em] \\ \end{array}$﻿

Method 3:

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

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

23) ﻿$5x^{2} + 17x + 6 = 0$﻿

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24) ﻿$6x^{2} + 19x = -10$﻿

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25) ﻿$8x^{2} - 33x + 4 = 0$﻿

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26) ﻿$8x^{2} - 26x = -21$﻿

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27) ﻿$10x^{2} + 37x = 36$﻿

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28) ﻿$12x^{2} + 20x - 77 = 0$﻿

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