Find the value

Question

Find the value of 8(a²+b²) if (a+b)² =225 and ab =7.5

Answer 1

(a + b)² = 225

a² + 2ab + b² = 225

 

Since ab = 7.5:

 

a² + 2(7.5) + b² = 225

 

Can you finish it from here?

Answer 2

Let’s start by finding two different expression for “a”

 

ab = 7.5 so a = 7.5/b

and 

(a+b)^2 = 225 

taking square roots on both sides, (a+b)=15

Solving for a, a = 15-b

 

To obtain a value for a, let’s equate these two expressions

7.5/b = 15-b

7.5 = 15b - b^2

Rearranging

b^2 - 15b + 7.5 = 0

From quadratic formula, b1=14.48 and b2=0.52. I’ll choose b2 as b. This choice doesn’t really matter since both equations (a+b) and (b1+b2) are equal to 15 

a+b = 15

a + 0.52 = 15

a = 15 - 0.52 = 14.48

 

Knowing a and b, let’s find the value of 8(a^2 + b^2)

a^2 = 14.48^2 = 209.67

b^2 = 0.52^2 = 0.2704

8*(209.67 + 0.2704) 

= 8*209.9404

= 1679.52