solve the quadratic equation x2-4x+7=0
Answers
Answer:
1
Use the quadratic formula
=
−
±
2
−
4
√
2
x=\frac{-{\color{#e8710a}{b}} \pm \sqrt{{\color{#e8710a}{b}}^{2}-4{\color{#c92786}{a}}{\color{#129eaf}{c}}}}{2{\color{#c92786}{a}}}
x=2a−b±b2−4ac
Once in standard form, identify a, b and c from the original equation and plug them into the quadratic formula.
2
−
4
+
7
=
0
x^{2}-4x+7=0
x2−4x+7=0
=
1
a={\color{#c92786}{1}}
a=1
=
−
4
b={\color{#e8710a}{-4}}
b=−4
=
7
c={\color{#129eaf}{7}}
c=7
=
−
(
−
4
)
±
(
−
4
)
2
−
4
⋅
1
⋅
7
√
2
⋅
1
x=\frac{-({\color{#e8710a}{-4}}) \pm \sqrt{({\color{#e8710a}{-4}})^{2}-4 \cdot {\color{#c92786}{1}} \cdot {\color{#129eaf}{7}}}}{2 \cdot {\color{#c92786}{1}}}
x=2⋅1−(−4)±(−4)2−4⋅1⋅7
2
Simplify
3
No real solutions because the discriminant is negative
Answer:
Step-by-step explanation:
x
=
2
±
√
3
i
Explanation:
Given:
x
2
−
4
x
+
7
=
0
While completing the square we will find that this takes the form of the sum of a square and a positive number. As a result it has no solution in real numbers, but we can solve it using complex numbers.
The imaginary unit
i
satisfies
i
2
=
−
1
The difference of squares identity can be written:
a
2
−
b
2
=
(
a
−
b
)
(
a
+
b
)
We can use this with
a
=
(
x
−
2
)
and
b
=
√
3
i
as follows:
0
=
x
2
−
4
x
+
7
0
=
x
2
−
4
x
+
4
+
3
0
=
(
x
−
2
)
2
+
(
√
3
)
2
0
=
(
x
−
2
)
2
−
(
√
3
i
)
2
0
=
(
(
x
−
2
)
−
√
3
i
)
(
(
x
−
2
)
+
√
3
i
)
0
=
(
x
−
2
−
√
3
i
)
(
x
−
2
+
√
3
i
)
Hence the two roots are:
x
=
2
+
√
3
i
and
x
=
2
−
√
3
i