Video: Forming and Solving Quadratic Equations

The side length of a square is 𝑥 cm, and the dimensions of a rectangle are 𝑥 cm and 2 cm. Given that the sum of their areas is 8 cm², determine the perimeter of the square.

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Video Transcript

The side length of a square is 𝑥 centimeters, and the dimensions of a rectangle are 𝑥 centimeters and two centimeters. Given that the sum of their areas is eight square centimeters, determine the perimeter of the square.

If we consider the two diagrams, the square with length 𝑥 centimeters and the rectangle with dimensions 𝑥 centimeters and two centimeters, we can calculate their areas by multiplying the length by the width. The area of the square is 𝑥 squared, as 𝑥 multiplied by 𝑥 is 𝑥 squared. The area of the rectangle is two 𝑥, as two multiplied by 𝑥 is two 𝑥. We were told in the question that the sum of their areas is eight square centimeters. This means that 𝑥 squared plus two 𝑥 is equal to eight. Subtracting eight from both sides of the equation gives us 𝑥 squared plus two 𝑥 minus eight equals zero.

As this quadratic equation is equal to zero, we can factorize it into two parentheses or brackets. Factorizing 𝑥 squared plus two 𝑥 minus eight gives us 𝑥 plus four multiplied by 𝑥 minus two, as four multiplied by negative two is negative eight. And four plus negative two is positive two.

Solving this quadratic equation gives us two possible values. Either 𝑥 plus four equals zero or 𝑥 minus two equals zero. We therefore have two values of 𝑥. Either 𝑥 equals negative four or 𝑥 equals positive two. As we are dealing with shapes, and in this case squares and rectangles, our answer must be 𝑥 equals two, as 𝑥 must have a positive value. A length must be positive.

As the length of the square is two centimeters, the perimeter can be calculated by multiplying two by four. Two multiplied by four is equal to eight. Therefore, the perimeter of the square is eight centimeters. It is worth noting in this question that the rectangle that was mentioned is also a square, as it too has dimensions two centimeters by two centimeters.

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