The grocery store parking lot is shaped like a parallelogram. Mr.Campbell is paving the parking lot.The area of the parking lot is 6x^2-3x-3m^2. what are the measures of the height and base of the parking lot.
i did 3(2x^2-x-m^2) as my first step but now i am confused.
That would be an answer, since all we need is two numbers or expressions which when multiplied give us 6x^2-3x-3m^2.
I have a suspicion that the middle term should be -3xm, then we could go further:
3(2x^2-x-m^2)
= 3(2x + m)(x - m)
but since we want only two factors, namely the base and height, the other options would be
(6x + 3m)(x - m) or (3x - 3m)(2x + m)
only true if you had the suspected misprint
To find the measures of the height and base of the parking lot, we need to factor the given expression, 6x^2 - 3x - 3m^2.
Step 1: Factor out the greatest common factor (GCF) of the terms in the expression, which is 3:
3(2x^2 - x - m^2)
Step 2: Factor the quadratic expression within the parentheses. We are looking for two binomials in the form (ax ± b)(cx ± d) whose product gives us 2x^2 - x - m^2.
The coefficient of x^2 is 2, so the factors must be in the form (2x ± ?)(x ± ?).
To find suitable values for the question marks, we need to consider factors of 2x^2 (-2x * -x or 2x * x) and factors of -m^2 (-m * m or m * -m) that can be combined to give -x in the middle term.
Step 3:
We take the factors of 2x^2, which are 2x and x, and the factors of -m^2, which are -m and m.
The possibilities for the binomial factors are:
(2x ± 1)(x ± 3m)
or
(2x ± m)(x ± 3)
Step 4: We multiply the binomials to check which one gives us the original expression:
(2x + m)(x - 3)
Therefore, the base of the parking lot is 2x + m, and the height is x - 3.
To find the measures of the height and base of the parking lot shaped like a parallelogram, we can use the area formula for a parallelogram.
The area of a parallelogram can be calculated by multiplying the base and the height. In this case, the area of the parking lot is given as 6x^2 - 3x - 3m^2.
So, we can set up the equation:
6x^2 - 3x - 3m^2 = base × height
Now, let's factor out 3 from the equation to make it easier to work with:
3(2x^2 - x - m^2) = base × height
Now, we need to determine the factors of the trinomial 2x^2 - x - m^2 to find the possible values for the base and height.
To find the factors, we can use methods like factoring, completing the square, or quadratic formula.
If we consider factoring, let's break down the trinomial as follows:
2x^2 - x - m^2 = (2x + p) (x + q)
Now, we multiply the coefficients 2 and -m^2 to get -2m^2. To find the factors of -2m^2 that add up to -1 (the coefficient of 'x'), we can try different combinations.
After some trial and error, we find that -2m^2 = -3m^2 + 2m^2
Therefore, the factors of the trinomial are:
2x^2 - x - m^2 = (2x - 3m^2) (x + 2m^2)
Now, we can rewrite the equation for the area of the parking lot using the factors:
3(2x - 3m^2) (x + 2m^2) = base × height
From this equation, the measure of the base is (2x - 3m^2), and the measure of the height is (x + 2m^2).