Math 96--Solving All Kinds of Equations--page 1
Name
A.
Linear (1 st Degree)--isolate the variable.
a.
5x + 9 = 13
b.
5x = 4
x=
2x ! 16 = 4x ! 30
2x ! 4x = !30 + 16
c.
4(3x + 2) = 7(5x + 9)
12x + 8 = 35x + 63
!2x = !14
12x ! 35x = 63 ! 8
x= 7
!23x = 55
x=
B.
Quadratic (2 nd Degree) That Factor--equation must equal zero to use factoring!
d.
x 2 + 7x = 0
e.
x 2 ! 3x = 28
x(x + 7) = 0
x 2 ! 3x ! 28 = 0
x = 0 or
x+ 7= 0
x 2 + 4x ! 7x ! 28 = 0
x = !7
x(x + 4) ! 7(x + 4) = 0
{!7, 0}
(x + 4)(x ! 7) = 0
x+ 4= 0
or
x!7= 0
x = !4
x= 7
{!4, 7}
f.
4x 2 = 5x + 6
4x 2 ! 5x ! 6 = 0
4x 2 ! 8x + 3x ! 6 = 0
4x(x ! 2) + 3(x ! 2) = 0
(4x + 3)(x ! 2) = 0
4x + 3 = 0
or
x!2= 0
4x = !3
x= 2
g.
x 2 = 81
x 2 ! 81 = 0
(x + 9)(x ! 9) = 0
x+ 9= 0
or
x = !9
{!9, 9}
x!9= 0
x= 9
x=
C.
Quadratic (2 nd Degree) That Don’t Factor–binomials that don’t factor use the square root property; trinomials that don’t
factor use completing the square or the quadratic formula. The results could be rational, irrational, or complex.
h.
x 2 ! 17 = 0
i.
x 2 + 12 = 0
j.
2x 2 ! 60 = 120
2
2
x = 17
x = !12
2x 2 = 180
x=
x=
x 2 = 90
x=
x=
x=
k.
(3x + 4) 2 = 10
l.
(2x ! 5) 2 = 16
3x + 4 =
2x ! 5 =
3x = !4
2x ! 5 = ± 4
x=
2x = 5 ± 4
2x = 5 + 4
2x = 9
or
or
2x = 5 ! 4
2x = 1
x=
or
x=
Math 96--Solving All Kinds of Equations--page 2
m.
x 2 + 8x ! 50 = 0
x 2 + 8x
= 50
n.
3x 2 ! 7x ! 5 = 0
a = 3, b = !7, c = !5
x 2 + 8x + 16 = 50 + 16
x=
(x + 4)(x + 4) = 66
x=
(x + 4) 2 = 66
x=
x+ 4=
x = !4 ±
D.
Equations with Fractions (Rational Expressions)--eliminate all denominators by multiplying all terms by the LCD.
Solve the new equation (linear–isolate the variable; quadratic–factor, use the square root property, or use the quadratic
formula). Check to make sure no result zeroes out any original denominator.
o.
p.
M ultiply all terms by LCD 20
5(2x !3) ! 2(4x ! 1) = 4(3)
10x ! 15 ! 8x + 2 = 12
2x ! 13 = 12
2x = 25
M ultiply all terms by LCD (x+ 3)(x!3). x … !3, 3
5(x ! 3) + 2(x + 3) = 7
5x ! 15 + 2x + 6 = 7
7x ! 9 = 7
7x = 16
x=
x=
q.
r.
M ultiply by LCD (x+ 7)(x!7). x … !7, 7
6(x ! 7) + 70 = 5(x + 7)
6x ! 42 + 70 = 5x + 35
6x + 28 = 5x + 35
6x ! 5x = 35 ! 28
x = 7; you can’t use 7 for a solution!
EM PTY SET!!
LCD (x)(x + 18). x … 0, !18.
3(x + 18) = x(x)
3x + 54 = x 2
0 = x 2 ! 3x ! 54
0 = (x + 6)(x ! 9)
x+ 6= 0
or
x!9= 0
x = !6 or
x= 9
{!6, 9}
s.
LCD (4)(x+ 5)(x !3); x … !5, 3.
4(x + 5)(x ! 3) + 6(4)(x ! 3) = 5(4)(x + 5)
4(x 2 ! 3x + 5x ! 15) + 24(x ! 3) = 20(x + 5)
4(x 2 + 2x ! 15) + 24x ! 72 = 20x + 100
4x 2 + 8x ! 60 + 24x ! 72 = 20x + 100
4x 2 + 32x ! 132 = 20x + 100
4x 2 + 32x ! 132 ! 20x ! 100 = 0
4x 2 + 12x ! 232 = 0
divide all terms by 4
x 2 + 3x ! 58 = 0
a = 1, b = 3, c = !58
x=
x=
so x =
Math 96--Solving All Kinds of Equations--page 3
E.
Radical Equations--isolate radical; remove radical (raise to a power to “ undo” the index); solve the new equation
(linear–isolate the variable; quadratic–factor, use the square root property, or use the quadratic formula); check (when the
original index is an even number).
t.
u.
x!4= 9
x = 13; check 9 = 9 so {13}
2x + 6 = 16
2x = 10
x = 5; this didn’t check 11 = 3 so EM PTY SET!!
v.
3x + 25 = x 2 + 14x + 49
0 = x 2 + 14x + 49 ! 3x ! 25
0 = x 2 + 11x + 24
(x + 3)(x + 8) = 0
x+ 3= 0
or
x+ 8= 0
x = !3
or
x = !8
{!3}
(x+ 7) 2 means (x+ 7)(x+ 7) Y x 2 + 7x + 7x + 49
x 2 + 3x + 8x + 24 Y x(x+ 3) + 8(x+ 3) Y (x+ 3)(x+ 8)
Now check in the original equation:
Check x = !3
Check x = !8
4 = 4, checks
1 = !1, doesn’t check
w.
2x = 216
x = 108 (you don’t have to check because it was a power of 3, but you can check) so {108}
F.
Checking your solution. You can always check your solution on any equation you solve. However, there are two types
of equations where you M UST check your solution. On any equation that has a denominator with variables, you must
check your solution to make sure your solution doesn’t zero out any original denominator. Also, on any equation that
contained an even index in the original equation, you must check your solution(s) in the original equation.
G.
Linear Inequalities. Isolate the variable; if you divide or multiply by a negative number, the inequality symbol reverses.
State result in interval notation.
x.
2x + 7 < 15
y.
!4m ! 7 # !31
2x < 8
!4m # !24
x < 4 so (!4, 4)
m $ 6 so [6, 4)
H.
3 rd or 4 th degree equations. M any 3 rd or 4 th degree equations use a combination of factoring and other methods from
above when they are solved.
z.
x 3 + 5x 2 + 12x + 60 = 0
aa.
x 4 ! 11x 2 + 28 = 0
2
x (x + 5) + 12(x + 5) = 0
x 4 ! 4x 2 ! 7x 2 + 28 = 0
2
(x + 5)(x + 12) = 0
x 2(x 2 ! 4) ! 7(x 2 ! 4) = 0
2
x+ 5= 0
or
x + 12 = 0
(x 2 ! 4)(x 2 ! 7) = 0
2
x = !5
or
x = !12
x2 ! 4 = 0
or
x2 ! 7 = 0
x=
x2 = 4
x=
x=
x= ± 2
or
x2 = 7
x=
Math 96--Solving All Kinds of Equations--page 4
I.
Substitution.
bb.
(x + 18) 2 + 9(x + 18) ! 22 = 0
a 2 + 9a ! 22 = 0
(a ! 2)(a + 11) = 0
a = 2 or a = !11
Then re-substitute a = x + 18
x + 18 = 2
or
x + 18 = !11
x = !16
or
x = !29
Let a = (x + 18)
You try these! Solve by an appropriate technique. Results may be rational, irrational, or complex.
1.
6x ! 9 = 4x ! 20
2.
x 2 + 8x = 0
3.
x 2 + 7x + 12 = 0
4
3x 2 + 10x ! 8 = 0
5.
x 2 = 20
6.
x 2 + 12 = 0
7.
x 2 + 6x + 10 = 0
8
2x 2 ! 7x ! 5 = 0
9.
10.
11.
12.
13.
14.
15.
18.
!6x + 18 < 66
21.
x 4 ! 7x 2 ! 18 = 0
16.
(x + 11) 2 = 36
17.
19.
x 3 + 6x 2 ! 3x ! 18 = 0
20.
22.
(x + 23) 2 + 10(x + 23) + 21 = 0
23.
(x ! 35) 2 ! 7(x ! 35) ! 18 = 0
24.
(2x + 5) 2 + 3(2x + 5) ! 40 = 0
x 3 + 7x 2 + 12x = 0
Answer Key. On some of the problems, I’ll show some of the work just to help you out.
1.
x=
2.
x(x + 8) = 0
3.
{0, !8}
4.
(3x ! 2)(x + 4) = 0
5.
square root property
x=
(x + 3)(x + 4) = 0
{!3, !4}
6.
square root property
x=
Math 96--Solving All Kinds of Equations--page 5
7.
quadratic formula or
8.
complete the square
quadratic formula
9.
LCD 20
10(7x+ 3) ! 4((5x ! 2) = 5(3)
x=
x = !3 ± i
x=
10.
LCD (x+ 9)(x!9); x … !9, 9
14 + 11(x ! 9) = 7(x + 9)
x = 37
11.
LCD (x+ 6)(x!6); x … !6, 6
7(x + 6) ! 3(x ! 6) = 84
x= 6
empty set
12.
LCD (4)(x); x … 0
x(x) = 4(x + 15)
x 2 ! 4x ! 60 = 0
{!6, 10}
13.
LCD (2)(x); x … 0
2(x + 4) = x(x)
0 = x 2 ! 2x ! 8
{!2, 4}
14.
isolate radical
cube both sides
x ! 3 = 64
x = 67
{67}
15.
square both sides
2x + 20 = x 2 + 12x + 36
0 = x 2 + 10x + 16
x = !2, ! 8; check
{!2}
16.
square root property
17.
LCD (4)(x)(x+ 7) x … 0, !7
18.
!6x < 48
x = !5 or x = !17
19.
x 2(x + 6) ! 3(x + 6) = 0
x > !8 so (!8, 4)
x=
20.
(x + 6)(x 2 ! 3) = 0
x(x 2 + 7x + 12) = 0
21.
(x 2 + 2)(x 2 ! 9) = 0
24.
Substitution; a = (2x + 5)
a 2 + 3a ! 40 = 0
a = 5 or a = !8
2x + 5 = 5 or 2x + 5 = !8
x(x + 3)(x + 4) = 0
{0, !3, !4}
22.
Substitution; a = (x + 23)
a 2 + 10a + 21 = 0
a = !3 or a = !7
x + 23 = !3 or x + 23 = !7
23.
Substitution; a = (x ! 35)
a 2 ! 7a ! 18 = 0
a = !2 or a = 9
x ! 35 = !2 or x ! 35 = 9
x = !26 or x = !30
x = 33 or x = 44
{!26, !30}
{33, 44}
x = 0 or x =