Let the width be x.
Length is 3 more than twice the width = 2x + 3
Area = x*(2x + 3)
x*(2x + 3) = 77
2x² + 3x = 77
2x² + 3x - 77 = 0 This is a quadratic equation. Solve by factorization.
Multiply first and last coefficients: 2*-77 = -154
We look for two numbers that multiply to give -154, and add to give the middle coefficient +3
Those two numbers are 14 and -11.
Check: 14*-11 = -154 14 + -11 = 14 - 11 = +3
We replace the middle term of +3x in the quadratic expression with 14x -11x
2x² + 3x - 77 = 0
2x² + 14x -11x - 77 = 0
2x(x + 7) - 11(x + 7) = 0
(2x - 11)(x + 7) = 0
2x - 11 = 0 or x + 7 = 0
2x = 0 + 11 x = 0 - 7
2x = 11 x = -7
x = 11/2 = 5.5
The solutions are x = -7 or 5.5.
Since we are solving for length of rectangle, x can't be negative. So we do away with x = -7.
x = 5.5 is the only valid solution.
Recall width = x = 5.5,
Length = 2x + 3 = 2*5.5 + 3 = 11 + 3 = 14
Therefore length = 14 yards, and width = 5.5 yards.