Question

If a+2b = 3 and ab =-5 then find the value of [1] a^2 + 4b^2 and [2] a^3 + 8b^3

Answer 1

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Answer 2

Concept

Using formulas

(a + b)² = a² + b² + 2ab

(a + b)³ = a³ + b³ + 3ab( a + b )

Given

1) a + 2b = 3

2) ab = -5

Find

1) a² + 4b²

2) a³ + 8b³

Solution

1) a² + 4b²

a + 2b= 3

Squaring both sides

(a + 2b)² = 9

a²  + 4b² + 4ab = 9

a² + 4b² + 4(-5) = 9

a² + 4b² = 9 + 20

a² + 4b² = 29

2) a³ + 8b³

a + 2b= 3

Cubing both sides

(a + 2b)³ = 27

a³ + 8b³ +3(a)(2b)(a + 2b) = 27

a³ + 8b³ +6(ab)(a + 2b) = 27

a³ + 8b³ +6(-5)(3) = 27

a³ + 8b³ -90 =27

a³ + 8b³ = 117

Hence the value of a² + 4b² = 29 and a³ + 8b³ = 117

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