solve this quadratic equation question?

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.

Answer 1

Radii of two circles:

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