Completing the Square (Part 3)

Illustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2

45 questions

Completing the Square (Part 3)

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Problem 1

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

a) ﻿$9x^{2} + 24x + 16 \quad \quad$﻿ b) ﻿$2x^{2} + 20x + 100 \quad \quad$﻿ c) ﻿$(7 - 3x)^{2} \quad \quad\quad$﻿ d) ﻿$(5x + 4)(5x - 4) \quad \quad$﻿ e) ﻿$(1 - 2x)(-2x + 1) \quad \quad\quad$﻿ f) ﻿$4x^{2} + 6x + \frac{9}{4} \quad \quad$﻿

Problem 2

Find the missing number that makes the expression a perfect square. Next, write the expression in factored form.

2) ﻿$49x^{2} - \underline{\quad}x + 16$﻿

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3) Write the expression in factored form.

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4) ﻿$36x^{2} + \underline{\quad}x + 4$﻿

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5) Write the expression in factored form.

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6) ﻿$4x^{2} - \underline{\quad}x + 25$﻿

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7) Write the expression in factored form.

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8) ﻿$9x^{2} + \underline{\quad}x + 9$﻿

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9) Write the expression in factored form.

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10) ﻿$121x^{2} + \underline{\quad}x + 9$﻿

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11) Write the expression in factored form.

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Problem 3

Find the missing number that makes the expression a perfect square. Next, write the expression in factored form.

12) ﻿$9x^{2} + 42x + \underline{\quad}$﻿

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13) Write the expression in factored form.

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14) ﻿$49x^{2} - 28x +\underline{\quad}$﻿

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15) Write the expression in factored form.

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16) ﻿$25x^{2} + 110x + \underline{\quad}$﻿

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17) Write the expression in factored form.

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18) ﻿$64x^{2} - 144x + \underline{\quad}$﻿

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19) Write the expression in factored form.

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20) ﻿$4x^{2} + 24x + \underline{\quad}$﻿

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21) Write the expression in factored form.

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Problem 4

Find the value of ﻿$c$﻿ to make the expression a perfect square. Then, write an equivalent expression in factored form.

22) Find the value of ﻿$c$﻿ to make ﻿$4x^2 \ + \ 4x$﻿ a perfect square ﻿$\left(ax^2 \ + \ bx \ + \ c\right)$﻿.

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

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24) Find the value of ﻿$c$﻿ to make ﻿$25x^2 \ - \ 30x$﻿ a perfect square ﻿$\left(ax^2 \ + \ bx \ + \ c\right)$﻿.

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

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Solve each equation by completing the square.

26) ﻿$4x^{2} + 4x = 3$﻿

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27) ﻿$25x^{2} - 30x + 8 = 0$﻿

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Problem 5

For each function f, decide if the equation f(x)=0 has 0, 1, or 2 solutions. Explain how you know.

28) For this function ﻿$f$﻿, decide if the equation ﻿$f(x) = 0$﻿ has 0, 1, or 2 solutions.

29) Explain how you know.

30) For this function ﻿$f$﻿, decide if the equation ﻿$f(x) = 0$﻿ has 0, 1, or 2 solutions.

31) Explain how you know.

32) For this function ﻿$f$﻿, decide if the equation ﻿$f(x) = 0$﻿ has 0, 1, or 2 solutions.

33) Explain how you know.

34) For this function ﻿$f$﻿, decide if the equation ﻿$f(x) = 0$﻿ has 0, 1, or 2 solutions.

35) Explain how you know.

36) For this function ﻿$f$﻿, decide if the equation ﻿$f(x) = 0$﻿ has 0, 1, or 2 solutions.

37) Explain how you know.

38) For this function ﻿$f$﻿, decide if the equation ﻿$f(x) = 0$﻿ has 0, 1, or 2 solutions.

39) Explain how you know.

Problem 6

Solve each equation.

40) ﻿$p^{2} + 10 = 7p$﻿

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41) ﻿$x^{2} + 11x + 27 = 3$﻿

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42) ﻿$(y + 2)(y + 6) = -3$﻿

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Problem 7

43) Which function could represent the height in meters of an object thrown upwards from a height of 25 meters above the ground t seconds after being launched?

a)

﻿$f$﻿(﻿$t$﻿)=-5﻿$t^{2}$﻿

b)

﻿$f$﻿(﻿$t$﻿)=-5﻿$t^{2}$﻿+25

c)

﻿$f$﻿(﻿$t$﻿)=-5﻿$t^{2}$﻿+25﻿$t$﻿+50

d)

﻿$f$﻿(﻿$t$﻿)=-5﻿$t^{2}$﻿+50﻿$t$﻿+25

Problem 8

A group of children are guessing the number of pebbles in a glass jar. The guesses and the guessing errors are plotted on a coordinate plane.

44) Which guess is furthest away from the actual number?

45) How far is the furthest guess away from the actual number?