.....
- ;;-.~
Chapter 7: Quadratic Equations and Graphs
Section 7.1
1. a. See Figure 2 on page 259 of the text. Note
that some data are rounded.
9. After entering the expressions in YI and Y2 and
setting the window variables, we can TRACE the
graph or use 2nd CALC intersect to find the
aDDroximatesolutions.
b. No
c. No, linear models have constant slope. The
graph of these data has changing slope.
3. Let s be the length of a side of the square.
A=S2
64 inches2 = S2
I~64 inches2 =..[;2
::1:8inches
= s
Ignore the negative solution; a side of the square
is 8 inches.
9 =x2
S. a.
I.J9 = 17
:!:3 = x
Il'IhrslQ:cti(lf)
M='t.OB2LtB29 1\'=5
The points of intersection are approximately
(4.08,5) and (4.08,5), so the solutions to the
equation 5
.3~ are x ::::::J: 4.08.
25 =S2
b.
!1
I..fi5 ..
::1:5 ".s
c.
·
11. ~
2x-3..0
2x=3
3
x = - =1.5
2
26 = S2
I.J26
=fl
I.J26=s
::I:5.1:::::s
The solutions for (a) and (b) are exact, but the
solutions for (c) are approximate; because 9 and
25 are perfect squares, but 26 is not.
7.
400 =1.2d2
400 = d2
1.2
1000
=d2
3
3=
o .
2X2
=3
x 2 =-3
2
x=::I:J%
Solving quadratic equations of the form
ax!- + C = 0 uses the same process as solving linear
equations, with the extra step of taking a square
root on both sides of the equation.
Use Math Frac feature
d
Thi
::I:~1000
::I: 18.26:::::
Dui!W~
2X2-3_0
.
slsexact
d
.
Th IS IS approximate b ecause
.
-1000
ISnot a
3
perfect square.
@ Houghton Mifflin Company. All rights reserved.
Section 7.1
13. Each of these quadratic equations can be solved
by factoring, see page 240 of the text for
review.
d.
105
x2-llx = -18
x2-11x+18=0
Diagonal product = 18x2
a. x2 + 4x + 3 = 0
Diagonal product
-2x, -9x is the factor pair of 1~ that adds to
-llx.
= 3x2
lx, 3x is the factor pair of 3~ that adds to 4x
(the middle tenn).
x2 + 4x + 3 = 0
x2 + Ix + 3x + 3 =0
(x2 + Ix) + (3x + 3)
(x2 -2x) + C9x+ 18) = 0
e. 2x2+ IOx+12= 0
or x=-3
b. x2 - 3x + 2 = 0
Diagonal product = 2x2
-lx, -2x is the factor pair of2x2 that adds to
-3x.
x2- 3x + 2 = 0
x2- Ix - 2x - 2 = 0
2(x2+Sx+6)=0
Diagonalproduct= 6x2
2x, 3x is the factor pair of ~ that adds to Sx.
2(x2 + 2x+ 3x+ 6)
2«x2
+ 2x) + (3x + 6))
2(x(x
+ 2) + 3(x + 2))
=0
=0
=0
2(x+2)(x+3)=0
2 = 0 (but 2 * 0) so x + 2 = 0 or x + 3 = 0
x=-2 or x=-3
(x2-1x)+(-2x+2)=0
x(x-l)+ -2(x-l) =0
(x-l)(x-2)=0
x -1 = 0 or x - 2 = 0
x=l or x..2
IS. When a quadratic equation has only an x2-tenn
and a constant. we can solve the equation by
isolating the x2-term and taking the square root on
both sides.
c. y2+Sy-14=0
Diagonal product = -14y2
-2y, 7y is the factor pair oCl4l
Sy.
y2+Sy-14=0
x2-2x-9x+18=0
x(x-2)+ -9(x-2)=0
(x-2)(x-9)=0
x-2=0
or x-9=0
x = 2 or x = 9
=0
x(x+l)+3(x+l)=0
(x+l)(x+3)=0
x+l=O or x+3=0
x=-1
x2 - llx + 18 =0
that adds to
a. 4x2- 49 = 0
4x2= 49
W
=:tJ49
2x =:t7
7
x=:t2
y2-2y+7y-14=0
(l-2y)+(7y-14)=0
y(y - 2) + 7(y - 2) = 0
(y - 2)(y + 7) = 0
y-2=0
or y+7=0
y = 2 or y = -7
b. Using the difference of two squares, see p 244
of the text for review
- 49 = 0
- 7)(2x + 7) = 0
4x2
(2x
2x - 7 =0 or 2x + 7 = 0
2x
= 7 or 2x = -7
7
-7
x=- orx= 2
2
c. The solutions from parts (b) and (c) are the
same.
@ Houghton Mifflin Company. All rights reserved.
106
Chapter 7: Quadratic Equations and Graphs
Skills and Review 7.1
17. We can solve this quadratic using either the
square root principle or the difference of two
squares.
Square Root Principle
x2 -64=0
x2 =64
x=:tJ64
x=:t8
Difference of Two Squares
x2 -64=0
(x-8)(x+8)=0
x - 8 = 0 or x + 8 = 0
x = 8 or x = -8
19. (2x - 3)2= (2x - 3)(2x- 3)
=4x2-6x-6x+9
=4x2-12x+9
21. a. x+ y=212
b. x
- Y = 83
c. Solve each equation for y to enter into the Y=
menu.
Yl = 'X + 212, Y2 = X - 83
Windows may vary: Xmin = '250,
Xmax = 250, Ymin = '250, Ymax = 250
Il'Jhrstcti';'(J
~=1Lt7.5
1\'=6'1.5
Graphical solutions are approximate:
x = 147.5 andy = 64.5.
The first number is 147.5 and the second
number is 64.5.
3
1
23. z--=-z+4 3
2
3 1
1
2
-z--=342
2
-z=- 5
3
4
15
z=8
z = 1.875
Check:
Substituting~
forz intothe originalequation
8
15 3 1 15 1
---=-*-+8 4 3 8 2
15 6 5 4
---=-+8 8 8 8
9 9
-=8 8
-C7) JC7)2-4*1*12
25. -+
2*1
2*1
7 ..149-48
=-+2
2
7 .Ji
=-+2 2
7 1
=-+2 2
8
2
=4