One side of a rectangle is 13 less than three times the length of the other side. The area of the rectangle is 480 square feet.
Solve the quadratic equation x (3x-13)=480 and use your solutions to fill in the blanks below.
How many solutions does the equation x (3x-13)=480 have ?
What is the length of the smaller side of the rectangle?
The quadratic equation x(3x-13) = 480 can be rewritten as 3x^2 - 13x - 480 = 0.
To find the solutions, we can use the quadratic formula:
x = [-(-13) ± √((-13)^2 - 4(3)(-480))] / 2(3)
x = [13 ± √(169 + 5760)] / 6
x = [13 ± √5929] / 6
x = [13 ± 77] / 6
So the two solutions are x = (13 + 77) / 6 = 90/6 = 15 and x = (13 - 77) / 6 = -64/6 = -10.67.
Since a negative length doesn't make sense in this context, we take x = 15 as the positive solution. The length of the smaller side of the rectangle is therefore 15 feet.
Therefore, the equation x(3x-13)=480 has 1 solution, and the length of the smaller side of the rectangle is 15 feet.