Add : -3x +4 and 2x^2-7x-2.
Answers
The answers are
x
=
1
2
and
x
=
−
4
.
Explanation:
First, factor the coefficient of
x
2
out of the first two terms to get
−
2
x
2
−
7
x
+
4
=
−
2
(
x
2
+
7
2
x
)
+
4
=
0
.
Next, take the coefficient of
x
inside the parentheses,
7
2
, divide it by 2 to get
7
4
, and then square that number to get
49
16
. Add this number inside the parentheses and then "balance" it by adding
−
2
⋅
49
16
on the other side of the equation to get
−
2
(
x
2
+
7
2
x
+
49
16
)
+
4
=
−
2
⋅
49
16
=
−
49
8
.
The reason this trick is a good idea is that the expression
x
2
+
7
2
x
+
49
16
is a perfect square. It equals
(
x
+
7
4
)
2
, so the equation becomes
−
2
(
x
+
7
4
)
2
+
4
=
−
49
8
, which is equivalent to
−
2
(
x
+
7
4
)
2
=
−
81
8
and
(
x
+
7
4
)
2
=
81
16
.
Now take the
±
square root of both sides to get
x
+
7
4
=
±
9
4
, leading to two solutions
x
=
9
4
−
7
4
=
2
4
=
1
2
and
x
=
−
9
4
−
7
4
=
−
16
4
=
−
4
.
You should check these in the original equation:
x
=
1
2
⇒
−
2
(
1
2
)
2
−
7
(
1
2
)
+
4
=
−
1
2
−
7
2
+
4
=
−
4
+
4
=
0
x
=
−
4
⇒
−
2
(
−
4
)
2
−
7
(
−
4
)
+
4
=
−
32
+
28
+
4
=
Answer:
( - 3x + 4 ) + ( 2x² - 7x - 2 )
-3x + 4 + 2x² - 7x - 2
2x² -10x -2