# solve this quadratic equation question?

Two circles have radii (x + 2) cm and 2x cm respectively. If the ratio of the area of the first circle to that of the second circle is (x + 7) : x 2, find the value of the radius of the smaller circle.

r1 = x + 2 cm

r2 = 2x cm

Given ratio of areas is A1 : A2 = ( x + 7 ) : x²

A1 = π(r1)² = π( x + 2 )²

A1 =  π( x² + 2x + 4 )

A2 = π(r2)² = π( 2x )²

A2 = 4πx²

Actual ratio of areas is A1 : A2 = π( x² + 2x + 4 ) : 4πx²

A1 : A2 = ( x² + 2x + 4 ) : 4x²

Given ratio of areas is equal to Actual ratio of areas

( x + 7 ) / x² = ( x² + 2x + 4 ) / 4x²

4( x + 7 ) = ( x² + 2x + 4 )

4x + 28 =  x² + 2x + 4

x² - 2x - 24 = 0

x² - 6x + 4x - 24 = 0

x(x - 6) + 4(x - 6) = 0

(x - 6)(x+ 4) = 0

By using zero product property : If AB = 0 then A = 0 , B = 0

(x - 6) = 0  and  (x+ 4) = 0

x = 6 and x = -4

x = -4 is invalid due to negative symbol.

x = 6 is valid.

Substitute x = 6 in radii.

r1 = x + 2 = 6+2 = 8

r2 = 2x = 2*6 = 12

The value of the radius of the smaller circle is 8 cm