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\[\sqrt{3}{x}^{2}+\sqrt{5}x=12\]
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−
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log
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sqrt(3)*x^2+sqrt(5)*x=12
sqrt(3)*x^2+sqrt(5)*x=12
\sqrt{3}{x}^{2}+\sqrt{5}x=12
1
Move all terms to one side.
3
x
2
+
5
x
−
12
=
0
\sqrt{3}{x}^{2}+\sqrt{5}x-12=0
3
x
2
+
5
x
−
1
2
=
0
2
Use the Quadratic Formula.
1
In general, given
a
x
2
+
b
x
+
c
=
0
a{x}^{2}+bx+c=0
a
x
2
+
b
x
+
c
=
0
, there exists two solutions where:
x
=
−
b
+
b
2
−
4
a
c
2
a
,
−
b
−
b
2
−
4
a
c
2
a
x=\frac{-b+\sqrt{{b}^{2}-4ac}}{2a},\frac{-b-\sqrt{{b}^{2}-4ac}}{2a}
x
=
2
a
−
b
+
b
2
−
4
a
c
,
2
a
−
b
−
b
2
−
4
a
c
2
In this case,
a
=
3
a=\sqrt{3}
a
=
3
,
b
=
5
b=\sqrt{5}
b
=
5
and
c
=
−
12
c=-12
c
=
−
1
2
.
x
=
−
5
+
5
2
−
4
3
×
−
12
2
3
,
−
5
−
5
2
−
4
3
×
−
12
2
3
{x}^{}=\frac{-\sqrt{5}+\sqrt{{\sqrt{5}}^{2}-4\sqrt{3}\times -12}}{2\sqrt{3}},\frac{-\sqrt{5}-\sqrt{{\sqrt{5}}^{2}-4\sqrt{3}\times -12}}{2\sqrt{3}}
x
=
2
3
−
5
+
5
2
−
4
3
×
−
1
2
,
2
3
−
5
−
5
2
−
4
3
×
−
1
2
3
Simplify.
x
=
−
5
+
5
+
48
3
2
3
,
−
5
−
5
+
48
3
2
3
x=\frac{-\sqrt{5}+\sqrt{5+48\sqrt{3}}}{2\sqrt{3}},\frac{-\sqrt{5}-\sqrt{5+48\sqrt{3}}}{2\sqrt{3}}
x
=
2
3
−
5
+
5
+
4
8
3
,
2
3
−
5
−
5
+
4
8
3
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x
=
−
5
+
5
+
48
3
2
3
,
−
5
−
5
+
48
3
2
3
x=\frac{-\sqrt{5}+\sqrt{5+48\sqrt{3}}}{2\sqrt{3}},\frac{-\sqrt{5}-\sqrt{5+48\sqrt{3}}}{2\sqrt{3}}
x
=
2
3
−
5
+
5
+
4
8
3
,
2
3
−
5
−
5
+
4
8
3
Done
Decimal Form: 2.064645, -3.355639
x=(-sqrt(5)+sqrt(5+48*sqrt(3)))/(2*sqrt(3)),(-sqrt(5)-sqrt(5+48*sqrt(3)))/(2*sqrt(3))
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