Algebra Review: Solving Quadratic Equations
Use With Lesson 7-2
Equations in the form
a
x
2
+
b
x
+
c
=
0
,
eh , x squared , plus b x plus , c equals 0 , comma where
a
≠
0
,
eh not equal to 0 comma are quadratic equations in standard form. You can solve some quadratic equations in standard form by factoring and using the Zero-Product Property:
If
a
b
=
0
,
eh b , equals 0 , comma then
a
=
0
eh equals 0 or
b
=
0
.
b equals 0 .
You can solve all quadratic equations in standard form using the quadratic formula:
If
a
x
2
+
b
x
+
c
=
0
,
eh , x squared , plus b x plus , c equals 0 , comma where
a
≠
0
,
eh not equal to 0 comma then
x
=
−
b
±
b
2
−
4
a
c
2
a
.
x equals . fraction negative b plus minus . square root of b squared , minus 4 eh c end root , over 2 eh end fraction . .
Example
Algebra Solve for x. For irrational solutions, give both the exact answer and the answer rounded to the nearest hundredth.
-
7
x
2
+
6
x
−
1
=
0
The equation is in standard form.
(
7
x
−
1
)
(
x
+
1
)
=
0
Factor.
7
x
−
1
=
0
or
x
+
1
=
0
Use the Zero-Product Property.
x
=
1
7
or
x
=
−
1
Solve for
x
table with 4 rows and 4 columns , row1 column 1 , 7 , x squared , plus , 6 x minus , 1 , column 2 equals , column 3 0 , column 4 cap the equation is in standard form. , row2 column 1 , open 7 x minus 1 . close , open x plus 1 , close , column 2 equals , column 3 0 , column 4 cap factor. , row3 column 1 , 7 x minus . 1 equals 0 , column 2 or , column 3 x plus . 1 equals 0 , column 4 cap use the cap zerominuscap product cap property. , row4 column 1 , x equals , 1 seventh , column 2 or , column 3 x equals , minus 1 , column 4 cap solve for . x , end table
-
−
3
x
2
−
5
x
+
1
=
0
The equation is in standard form.
a
=
−
3
,
b
=
−
5
,
c
=
1
Identify
a
,
b
,
and
c
.
x
=
−
(
−
5
)
±
(
−
5
)
2
−
4
(
−
3
)
(
1
)
2
(
−
3
)
Substitute in the quadratic formula
.
x
=
5
±
37
6
Simplify
.
x
=
5
±
37
6
or
x
=
−
5
−
37
6
Write the two solutions separately
.
x
≈
−
1.85
or
x
≈
0.18
Use a calculator and round to the nearest hundredth
.
table with 6 rows and 3 columns , row1 column 1 , negative 3 , x squared , minus 5 x plus 1 , column 2 equals 0 , column 3 cap the equation is in standard form. , row2 column 1 , eh , column 2 equals negative 3 comma b equals negative 5 comma c equals 1 , column 3 cap identify . eh comma b comma , and , c . , row3 column 1 , x , column 2 equals . fraction negative . open , negative 5 , close . plus minus . square root of open , negative 5 , close squared . minus 4 . open , negative 3 , close . open 1 close end root , over 2 . open , negative 3 , close end fraction , column 3 cap substituteinthequadraticformula . . , row4 column 1 , x , column 2 equals . fraction 5 plus minus square root of 37 , over 6 end fraction , column 3 cap simplify , . , row5 column 1 , x , column 2 equals . fraction 5 plus minus square root of 37 , over 6 end fraction . or , x equals negative . fraction 5 minus square root of 37 , over 6 end fraction , column 3 cap writethetwosolutionsseparately . . , row6 column 1 , x , column 2 almost equal to negative , 1.85 , or , x almost equal to , 0.18 , column 3 cap useacalculatorandroundtothenearesthundredth . . , end table
Exercises
Algebra Solve for x. For irrational solutions, give both the exact answer and the answer rounded to the nearest hundredth.
-
x
2
+
5
x
−
14
=
0
x squared , plus , 5 x minus . 14 equals 0
-
4
x
2
−
13
x
+
3
=
0
4 , x squared , minus , 13 x plus . 3 equals 0
-
2
x
2
+
7
x
+
3
=
0
2 , x squared , plus , 7 x plus . 3 equals 0
-
5
x
2
+
2
x
−
2
=
0
5 , x squared , plus , 2 x minus . 2 equals 0
-
2
x
2
−
10
x
+
11
=
0
2 , x squared , minus , 10 x plus . 11 equals 0
-
8
x
2
−
2
x
−
3
=
0
8 , x squared , minus , 2 x minus . 3 equals 0
-
2
x
2
+
3
x
−
20
=
0
2 , x squared , plus , 3 x minus . 20 equals 0
-
x
2
−
x
−
210
=
0
x squared , minus , x minus . 210 equals 0
-
x
2
−
4
x
=
0
x squared , minus , 4 x equals 0
-
x
2
−
25
=
0
x squared , minus , 25 equals 0
-
6
x
2
+
10
x
=
5
6 , x squared , plus . 10 x equals 5
-
1
=
2
x
2
−
6
x
1 equals . 2 , x squared . negative 6 x
