Answer:
Step-by-step explanation:
(4x−1)
2
=(4x−1)(4x−1)=16x
2
−8x+1
(
4
�
+
11
)
2
=
(
4
�
+
11
)
(
4
�
+
11
)
=
16
�
2
+
88
�
+
121
(4x+11)
2
=(4x+11)(4x+11)=16x
2
+88x+121
Now, substitute these expressions back into the equation:
�
2
+
(
16
�
2
−
8
�
+
1
)
=
(
16
�
2
+
88
�
+
121
)
x
2
+(16x
2
−8x+1)=(16x
2
+88x+121)
Expanding further:
�
2
+
16
�
2
−
8
�
+
1
=
16
�
2
+
88
�
+
121
x
2
+16x
2
−8x+1=16x
2
+88x+121
Combine like terms:
17
�
2
−
8
�
+
1
=
16
�
2
+
88
�
+
121
17x
2
−8x+1=16x
2
+88x+121
Now, let's subtract
16
�
2
+
88
�
+
121
16x
2
+88x+121 from both sides:
17
�
2
−
16
�
2
−
8
�
−
88
�
+
1
−
121
=
0
17x
2
−16x
2
−8x−88x+1−121=0
�
2
−
96
�
−
120
=
0
x
2
−96x−120=0
Now, we have a quadratic equation. To solve it, we can use the quadratic formula:
�
=
−
�
±
�
2
−
4
�
�
2
�
x=
2a
−b±
b
2
−4ac
Where
�
=
1
a=1,
�
=
−
96
b=−96, and
�
=
−
120
c=−120.
Plugging these values into the quadratic formula:
�
=
−
(
−
96
)
±
(
−
96
)
2
−
4
(
1
)
(
−
120
)
2
(
1
)
x=
2(1)
−(−96)±
(−96)
2
−4(1)(−120)
�
=
96
±
9216
+
480
2
x=
2
96±
9216+480
�
=
96
±
9696
2
x=
2
96±
9696
�
=
96
±
98.47
2
x=
2
96±98.47
Now, we have two possible solutions:
�
=
96
+
98.47
2
x=
2
96+98.47
�
=
96
−
98.47
2
x=
2
96−98.47
Solving these:
�
=
194.47
2
≈
97.235
x=
2
194.47
≈97.235
�
=
−
2.47
2
≈
−
1.235
x=
2
−2.47
≈−1.235
So, the solutions to the equation are approximately
�
≈
97.235
x≈97.235 and
�
≈
−
1.235
x≈−1.235.
