if x and y are ywo real numbers 9x^2+4y^2=97 and xy =97

Question

Then find value of 27x^3+64y^3"?

Answer 1

 

 The equations are  9x²+4y² = 97 and xy =97 .

Rewrite the equations

(3x)² + (2y)² = 97  and  xy = 97

The above given equation look like a² + b²

ab = (3x)(2y) = 6xy = 6(97) = 582

We have to calculate  27x³+64y³  ⇒ (3x)³ + (4y)³ .

The above  equation look like a³ + b³ 

a³ + b³  = (a + b)(a²− ab + b²)

(a+b)² = a² + b² +2ab

(a+b)² = 97 + 2(582)

(a+b)² = 1261

a+b = √(1261)

a²− ab + b² = 97 - 582 

a²− ab + b²  = -485

a³ + b³  = (a + b)(a²− ab + b²)

a³ + b³  = √(1261)(-485)

a³ + b³  = -485√(1261)

So   27x³+64y³ = -485√(1261) .