Answer:
Length = 14 in.
Width = 7 in.
Explanation:
The formula for the area of a rectangle is given by:
A = lw, where
- A is the area in units squared,
- l is the length,
- and w is the width.
Since the width is 7 in. less than the length, we can show this with the following equation:
w = l - 7
Now we can substitute l - 7 for w in the area formula and 98 for A to find l, the length of the rectangle:
Step 1: Distribute the l:
98 = l(l - 7)
98 = l^2 - 7l
Step 2: Subtract l^2 from both sides:
We have a quadratic and in order to solve, we'll need to put it in standard form, whose general equation is given by:
ax^2 + bx + c = 0
Thus, we'll need to move l^2 and -7l to the left-hand side of the equation by first subtracting l^2 from both sides:
(98 = l^2 - 7l) - l^2
-l^2 + 98 = -7l
Step 3: Add 7l to both sides to put the quadratic in standard form:
Now we can add 7l to both sides to put the quadratic in standard form:
(-l^2 + 98 = -7l) + 7l
-l^2 + 7l + 98 = 0
Step 4: Factor the quadratic:
- Since the quadratic is in standard form, -1 is our a value, 7 is our b value, and 98 is our c value.
- We can factor the quadratic by finding two numbers whose product equals a * c (-1 * 98) and whose sum is b (7).
- -7 and 14 meets these two requirements as -7 * 14 = 98 and -7 + 14 = 7.
- When putting the quadratic in factored form, we use the opposite sign of -7 and 14.
Thus, the quadratic in factored form is (l + 7)(l - 14) = 0
Step 5: Solve the quadratic by setting each term equal to 0:
- Since our variable is l, solving the quadratic will give us the length of the rectangle.
We can solve the quadratic by setting (l + 7) and (l - 14) equal to 0:
Setting (l + 7) equal to 0:
(l + 7 = 0) - 7
l = -7
Setting (l - 14) equal to 0:
(l - 14 = 0) + 14
l = 14
Because we can't have a negative side length, the length is 14 in.
Step 6: Identify the width:
- The only number which you can multiply with 14 to get 98 is 7.
Thus, the width is 7 in.