If 2x=3+√5 and y=x^3 then y satisfies the quadratic equation (i) y^2 -18y+1
Answers
Answer with Step-by-step explanation:
Quadratic equation :
Substitute the values then we get
Using identity:
Using identity:
LHS=RHS
Hence,y satisfies the quadratic equation.
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https://brainly.in/question/12770535:Answered by Spiderman
Answer:
x=
2
3+
5
y=x^3y=x
3
Quadratic equation :y^2-18y+1y
2
−18y+1
Substitute the values then we get
(x^3)^2-18x^3+1(x
3
)
2
−18x
3
+1
x^6-18x^3+1x
6
−18x
3
+1
x^3(x^3-18)+1x
3
(x
3
−18)+1
(\frac{3}{2}+\frac{\sqrt 5}{2})^3((\frac{3}{2}+\frac{\sqrt 5}{2})^3-18)+1(
2
3
+
2
5
)
3
((
2
3
+
2
5
)
3
−18)+1
Using identity:(a+b)^3=a^3+b^3+3a^2b+3ab^2(a+b)
3
=a
3
+b
3
+3a
2
b+3ab
2
(\frac{27}{8}+\frac{5\sqrt 5}{8}+\frac{27\sqrt 5}{8}+\frac{45}{8})((\frac{27}{8}+\frac{5\sqrt 5}{8}+\frac{27\sqrt 5}{8}+\frac{45}{8}-18)+1(
8
27
+
8
5
5
+
8
27
5
+
8
45
)((
8
27
+
8
5
5
+
8
27
5
+
8
45
−18)+1
(\frac{27+45}{8}+\frac{27\sqrt 5+5\sqrt 5}{8})(\frac{27+45}{8}+\frac{27\sqrt 5+5\sqrt 5}{8}-18)+1(
8
27+45
+
8
27
5
+5
5
)(
8
27+45
+
8
27
5
+5
5
−18)+1
(9+\frac{32\sqrt 5}{8})(9+\frac{32\sqrt 5}{8}-18)+1(9+
8
32
5
)(9+
8
32
5
x= 3+√5/2 & y=x³, then y satisfies the quadratic equation
−18)+1
(9+\frac{32\sqrt 5}{8})(\frac{32\sqrt 5}{8}-9)+1(9+
8
32
5
)(
8
32
5
−9)+1
(\frac{32\sqrt 5}{8})^2-9^2+1(
8
32
5
)
2
−9
2
+1
Using identity:(a+b)(a-b)=a^2-b^2(a+b)(a−b)=a
2
−b
2
80-81+1=080−81+1=0
LHS=RHS