Solving the quadratic equation derived from the given proportion yields two solutions: x = -9 or x = 6. These values satisfy the original equation (x + 3)/(x + 4) = 2/(x + 18).
To solve for x in the given equation (x + 3)/(x + 4) = 2/(x + 18), you can follow these steps:
Set up a proportion using the corresponding sides of the two similar triangles:
(x + 3)/(x + 4) = 2/(x + 18)
Cross-multiply to eliminate the fractions:
(x + 3)(x + 18) = (x + 4)(2)
Expand both sides and bring terms to one side of the equation:
x^2 + 21x - 54 = 0
Solve the quadratic equation by factoring, completing the square, or using the quadratic formula.
Factoring:
(x - 6)(x + 9) = 0
Setting each factor to zero and solving for x:
x - 6 = 0 ⟹ x = 6
x + 9 = 0 ⟹ x = -9
So, the solutions to the equation are x = -9 or x = 6.