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5-59. BACKYARD SOLUTIONS

a.
b.
Draw a diagram and write an equation to represent each scenario below. Solve the
equation, and then explain whether the solutions make sense.
Ernie is going to install a square
(x + 3)2 = 169
hot tub in his backyard. He plans
|x + 3| = 13
to add a 3-foot-wide deck on
x + 3 = 13 or x + 3 = -13
two adjacent sides of the hot
x = 10 or x = -16
tub. If Ernie’s backyard (which is
10 feet x 10 feet hot tub
also a square) has 169 square feet
of space, what is the largest hot
tub he can order?
(x + 1)2 = 12
Gabi is creating a decorative rock
|x + 1| = 12
garden that will cover 12 square
Rock .5 ft x + 1 =2 3 or x + 1 = -2 3
feet of her back yard, including its .5 ft
garden
frame. She plans to build a 6-inch
x = -1+ 2 3 or x = -1- 2 3
wide wood frame around the
x = 2.5
or
x = -4.5
Wood
frame
rock garden to keep the rocks in
2.5 feet x 2.5 feet rock garden
place. If the framed rock garden
is square, how much area will
Gabi need to cover with
decorative rocks?
5.2.1 Perfect Square Equations
January 25, 2016
Objectives
 CO: SWBAT
solve quadratic equations in
perfect square form and express their
solutions in exact and approximate forms.
 LO: SWBAT
explain how many solutions
a quadratic equation written in perfect
square form has.
5-60. In problem 5-59, did you notice anything special about the forms of the
equations you wrote? Did you use the Zero Product Property to solve your
equations, or were you able to solve them a different way? Without rewriting,
determine the value(s) of x that make(s) each equation true.
a. (x – 1)2 = 4
b. (x – 1)2 = 0
c. (x – 1)2 = –4
|x – 1| = 2
|x – 1| = 0
x – 1 = 2 or x – 1 = -2 x – 1 = 0
x=3
or x = -1
x=1
d. What method did you use
No solution because no
number times itself will
ever give a negative.
to solve each equation?
5-61. The quadratic equation (x – 3)2 = 12 is written in perfect
square form. It is called this because the expression (x – 3)2
forms a square when built with algebra tiles.
a.
Solve (x – 3)2 = 12. Write your answer in exact
form (or radical form). That is, write it in a
form that is precise and does not have any
rounded decimals.
(x – 3)2 = 12
|x – 3| = 12
x – 3 = 2 3 or x – 3 = - 2 3
x = 3 + 2 3 or x =3 - 2 3
How many solutions are there?
b.
2
c.
The solution(s) from part (a) are irrational.That
is, they are decimals that never repeat and never
end. Write the solution(s) in approximate
decimal form. Round your answers to the
nearest hundredth.
x ≈ 6.46 or –0.46
5-62. THE NUMBER OF SOLUTIONS

The equation in problem 5-61 had two solutions. However, in problem 5-60, you saw that a
quadratic equation might have one solution or no solutions at all. How can you determine
the number of solutions to a quadratic equation?
With your team, solve the equations below. Express your answers in both exact form
and approximate form. Look for patterns among the equations with no solution and
those with only one solution. Be ready to report your patterns to the class.
a.
(2x – 3)2 = 49
x = –2 or 5
2 solutions
b.
(7x – 5)2 = –2
No solution
c.
d.
(5 – 10x)2 = 0
x=½
1 solution
e.
(x + 2)2 = –10
No solution
f.
2 solutions → squared equals positive
1 solution → squared equals zero
No solution → squared equals negative
(x + 4)2 = 20
x = 3 + 2 3 or 3 - 2 3
x ≈ 0.47 or –8.37
2 solutions
(x + 11)2 + 5 = 5
x = -11
1 solution