The are of a rectangular barnyard is given by the trinomial 12x^2+14x-40. What are the possible dimensions of the barnyard? Use factoring.
To find the possible dimensions of the barnyard, we need to factor the given trinomial.
We can start by looking for common factors of the coefficients. The greatest common factor of 12, 14, and 40 is 2, so we can factor out a 2 from each term:
12x^2 + 14x - 40 = 2(6x^2 + 7x - 20)
Now we need to factor the quadratic expression inside the parentheses. We can use the factoring method of finding two numbers whose product is equal to the product of the first and last coefficients, and whose sum is equal to the middle coefficient. In this case, those numbers are 10 and -2, because:
10 x -2 = -20
10 + (-2) = 7
So we can write:
6x^2 + 7x - 20 = (2x - 5)(3x + 4)
Substituting this back into the original expression, we get:
2(2x - 5)(3x + 4)
This represents the area of the barnyard. The dimensions must be such that the product of two numbers is equal to this expression, and the sum of those two numbers is equal to the coefficient of the x-term (in this case, 14).
So the possible dimensions are:
2x - 5 = 0 and 3x + 4 = 0
or
2x - 5 = 2(3x + 4) and 3x + 4 = 2x - 5
Solving these equations, we get:
x = 5/2 and x = -4/5
or
x = -14 and x = 9
Since the dimensions of the barnyard must be positive, we can discard the negative values of x.
Therefore, the possible dimensions of the barnyard are:
5/2 by 24, or
9 by 20.
To find the possible dimensions of the barnyard, we need to factor the trinomial 12x^2 + 14x - 40.
Step 1: Factor out the greatest common factor (GCF), if any.
In this case, the GCF of the coefficients of 12x^2, 14x, and -40 is 2. So, we can factor out 2 from each term to simplify the expression:
2(6x^2 + 7x - 20)
Step 2: Factor the remaining quadratic expression inside the parentheses.
We need to find two numbers whose product is equal to the product of the leading coefficient (6) and the constant term (-20), and whose sum is equal to the coefficient of the middle term (+7).
The numbers that satisfy these conditions are +10 and -2, because (10 * -2 = -20) and (10 + -2 = 7).
So, we can rewrite the quadratic expression as:
2(6x^2 + 10x - 2x - 20)
Step 3: Group and factor by grouping.
Group the terms as follows:
2[(6x^2 + 10x) + (-2x - 20)]
Now, factor out the greatest common factor from each grouping:
2[2x(3x + 5) - 2(3x + 5)]
Step 4: Factor out the common binomial.
We can factor out the binomial (3x + 5):
2(3x + 5)(2x - 1)
So, the factored form of the trinomial 12x^2 + 14x - 40 is 2(3x + 5)(2x - 1).
From this, we can determine the possible dimensions of the barnyard:
- Length: (3x + 5)
- Width: (2x - 1)
The barnyard could have dimensions of (3x + 5) by (2x - 1), where x can be any real number.