Question

Two squares have sides x cm and (x + 5) cm. The sum of their areas is 697 sq. cm.

1. Express this as an algebraic equation in x. 
2. Solve this equation to find the sides of the squares.

Answer 1

i. Area of the first square $$= x^2\text{ cm}^2$$. 

Area of the second square $$= (x + 5)^2\text{ cm}^2$$. 

Sum of their areas: $$x^2 + (x + 5)^2 = 697$$

$$x^2 + x^2 + 10x + 25 = 697$$

$$2x^2 + 10x - 672 = 0$$ 

Dividing throughout by 2: $$x^2 + 5x - 336 = 0$$ 

ii. Factoring the equation: $$x^2 + 21x - 16x - 336 = 0$$ 

$$x(x + 21) - 16(x + 21) = 0$$

$$(x + 21)(x - 16) = 0$$

$$x = -21 \quad \text{or} \quad x = 16$$ 

Since side length cannot be negative, reject $$x = -21$$, so $$x = 16\ \text{cm}$$.

Side of the first square $$= 16\ \text{cm}$$. 

Side of the second square $$= 16 + 5 = 21\ \text{cm}$$.