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Answer:
To solve the equation
3
x
(
x
−
2
)
=
7
3x(x−2)=7, follow these steps:
Distribute
3
x
3x inside the parentheses:
3
x
2
−
6
x
=
7
3x
2
−6x=7
Move all terms to one side of the equation to set it to zero:
3
x
2
−
6
x
−
7
=
0
3x
2
−6x−7=0
Solve the quadratic equation using the quadratic formula,
x
=
−
b
±
b
2
−
4
a
c
2
a
x=
2a
−b±
b
2
−4ac
, where
a
=
3
a=3,
b
=
−
6
b=−6, and
c
=
−
7
c=−7:
Calculate the discriminant:
Δ
=
b
2
−
4
a
c
=
(
−
6
)
2
−
4
⋅
3
⋅
(
−
7
)
=
36
+
84
=
120
Δ=b
2
−4ac=(−6)
2
−4⋅3⋅(−7)=36+84=120
Compute the square root of the discriminant:
120
≈
10.95
120
≈10.95
Apply the quadratic formula:
x
=
−
(
−
6
)
±
120
2
⋅
3
=
6
±
10.95
6
x=
2⋅3
−(−6)±
120
=
6
6±10.95
Calculate the two solutions:
x
1
=
6
+
10.95
6
≈
16.95
6
≈
2.825
x
1
=
6
6+10.95
≈
6
16.95
≈2.825
x
2
=
6
−
10.95
6
≈
−
4.95
6
≈
−
0.825
x
2
=
6
6−10.95
≈
6
−4.95
≈−0.825
So, the solutions are approximately
x
≈
2.825
x≈2.825 and
x
≈
−
0.825
x≈−0.825.