Quadratic Equations Applications

Question

The dimensions of a square are altered so that one dimension is increased by 7 feet, while the other is decreased by 2 feet. The area of the resulting rectangle is 90 ft². Find the original area of the square.

Answer 1

Hi Jordan,

The most straightforward way to solve this type of question would be to write out the formula for the area of a square.

x² = A₁ where x is the length of one side of the square

Now when you change the dimensions, the new formula becomes

(x+7)(x-2) = 90 ft² = A₂ because one pair of sides is increased by 7 ft and the other is decreased by 2 ft

In order to find the area of the original square (A₁), you first have to solve for x. We cannot do that with the first equation, but we can do that with the second equation: (x+7)(x-2) = 90.

Step 1: Multiply the right side of the equation using FOIL to get x² - 2x + 7x -14 which can be simplified to x² + 5x - 14.

Step 2: Move the 90 over to the left side so you have 0 on the right side of the equation:

x² + 5x - 14 - 90 = 0 then becomes x² + 5x - 104 = 0

Step 3: Now we need to find which two numbers, when multiplied, equal -104, but when added together, they make 5. This can take a bit of trial and error. You would want to list all of the factors of 104:

2, 52

4, 26

8, 13 ... ok, so the difference between 8 and 13 is 5, so we have a winner!

Step 4: Now we can factor the left side of the equation to be: (x+13)(x-8) You can check this by using FOIL to see that we get x² - 8x + 13x - 104 which equals x² + 5x - 104 which is what we want.

Step 5: Now that we have factored our quadratic equation, we see that x can be either -13 or 8. Because -13 does not make sense in the context of this problem (the side of a square cannot be a negative distance), the answer is x = 8.

Step 6: Now that we know the original sides of the square are 8 units long, we go back to our first equation and plug it in: 8² = 64 ft² and that is the area of the original square.

Answer 2

Let x represent one side of the square.

Your new shape will have dimensions (x + 7) by (x - 2)

The area of this new shape is 90 ft², so

A = b*h

90 ft² = (x + 7)*(x - 2) ft² (I am going to omit units for the remainder, everything is in ft²)

90 = x² + 5x - 14

x² + 5x - 104 = 0

(x + 13)*(x - 8) = 0

x = -13, 8

Since x cannot be negative (x is a side of a square), we are left with x = 8. Additionally, since we factored the equation into two terms, each term is in ft, so x is in ft.

With the side being 8 ft, the original area is 8 ft * 8 ft = 64 ft²

Let's check our work:

Original side is 8 ft.

Modified sides are therefore 8 + 7 = 15 and 8 - 2 = 6.

Area of this modified shape is then 15 ft * 6 ft = 90 ft² (what we expect)