For this problem, we know the area and the perimeter of a rectangle. We make equations, and substitute so that everything is in terms of one variable.

We then multiply the brackets, and work towards having 0 on one side, and an algebraic expression on the other. That expression is factored (using the area model in the box) and the factors are written with the product=0.

We then find the roots by setting each bracket equal to 0.

Once we get the 2 values for “b”, we use each one to calculate a value for “a”.

This question relates the legs of the triangle to the hypotenuse using algebra. One leg is 7cm longer than the other, and the hypotenuse is 2 cm more than the shortest leg.

We use the pythagorean theorem to create our equation. We need to expand by multiplying the binomial by itself to create a trinomial. We simplify our equation to get 0 on one side.

We then factor (using the box) and we write our factors in brackets equal to 0. We now find the roots by setting each bracket equal to 0 and solving for x.

We need to check at the end if the x value is acceptable. We know that side lengths must be positive, so x must be 15, and not 3.