The length of a rectangle is two centimetres more than its width. If its area is 80 square centimetres, what are its length and width?
If we let the width be 𝑥 centimetres, then the length of the rectangle will be 𝑥 plus two as it is two centimetres longer than the width. The area of the rectangle is 80 square centimetres. Therefore, 𝑥 multiplied by 𝑥 plus two must be equal to 80, as the area of a rectangle is calculated by multiplying the length by the width. Writing this as an equation gives us 𝑥 multiplied by the parentheses 𝑥 plus two equals 80.
Expanding or multiplying out the parentheses gives us 𝑥 squared plus two 𝑥, as 𝑥 multiplied by 𝑥 is 𝑥 squared and 𝑥 multiplied by two is two 𝑥. In order to solve this quadratic equation, we must ensure that it is equal to zero. Subtracting 80 from both sides leaves us with 𝑥 squared plus two 𝑥 minus 80 equals zero.
Factorizing this quadratic gives us 𝑥 minus eight multiplied by 𝑥 plus 10. This is because negative eight multiplied by 10 is negative 80. And negative eight plus 10 is positive two. These are the free term and the coefficient of 𝑥, respectively. The free term is negative 80 and the coefficient of 𝑥 is positive two.
Solving this equation gives us two values: 𝑥 equals eight or 𝑥 equals negative 10. However, as we are dealing with a rectangle and length, our answer must be positive. Therefore, the answer for 𝑥 is 𝑥 equal to eight. As the width is now equal to eight centimetres, we can work out the length which, was two centimetres more than the width.
This means that the length of the rectangle is 10 centimetres. Multiplying eight centimetres by 10 centimetres gives us 80 square centimetres. As this was the area of the rectangle, we know that our answers the width equals eight centimetres and the length equals 10 centimetres are correct.