The area of a rectangular room is given by the trinomial x^2+7x-30. What are the possible dimensions of the rectangle? Use factoring. (1 point)
(x+10)(x-3)
Answer:
(3x + 5) and (x - 6).
Step-by-step explanation:
3x^2-13x-30
Use the 'ac' method:
a * c = 3*-30 = -90.
We need two numbers whose product is -90 and whose sum is -13. They are -18 and + 5, so we have:
3x^2 - 18x + 5x - 30
Factor by grouping:
= 3x(x - 6) + 5(x - 6)
= (3x + 5)(x - 6).
My answer was from another version of the question on Unit 3 Lesson 9: Polynomials and Factoring Unit Test Part 1 on Connexus, sorry lol
Well, let's solve this trinomial puzzle and find the dimensions of the rectangular room in a hilarious way, shall we?
To factor the trinomial x^2+7x-30, we need to find two numbers whose product is -30 (the last term) and whose sum is 7 (the coefficient of the middle term).
Now, -30, let's face it, that's a lot of numbers to try out, like finding a needle in a haystack! But hey, let's juggle with some numbers and see what we get.🤹♂️
Hmm, how about -5 and 6? If we multiply them, we get -30, and if we add them, we get 1! Oops, not exactly 7, but hey, we're getting close. Let's keep cracking up and trying other numbers.🤡
Ding ding ding! We've got a winner! The numbers -3 and 10, when multiplied, give us -30, and when added, give us 7! Jackpot!🥳
So, the possible dimensions of the rectangle are (x-3) and (x+10). If we set both expressions equal to zero, we can solve for x and get the specific dimensions of the rectangle. But remember, this is all just a playful little math game!😜
To find the possible dimensions of the rectangle, we need to factor the given trinomial: x^2 + 7x - 30.
Step 1: Write the trinomial in its factored form.
The factored form of the trinomial will have the form: (x + m)(x + n), where 'm' and 'n' are the possible factors of the constant term (-30) that add up to the coefficient of the linear term (+7).
Step 2: Determine the factors of the constant term.
The constant term in this case is -30. To find the factors, we need to find pairs of numbers that multiply to -30.
The factor pairs for -30 are: (1, -30), (-1, 30), (2, -15), (-2, 15), (3, -10), (-3, 10), (5, -6), and (-5, 6).
Step 3: Find the factors that add up to the coefficient of the linear term.
In this case, the coefficient of the linear term is +7. From the factor pairs above, the pair that adds up to 7 is (2, -15).
Step 4: Write the trinomial in its factored form.
Using the pair (2, -15), we can write the factored form as: (x + 2)(x - 15).
Step 5: Determine the possible dimensions of the rectangle.
Since the area of a rectangle is given by the product of its length and width, the possible dimensions of the rectangle are the values inside the parentheses: (x + 2) and (x - 15).
Therefore, the possible dimensions of the rectangle are: (x + 2) and (x - 15).
x^2 + 7x - 30
= (x+10)(x-3)
so the sides could be (x+10) and (x-3) , were x > 3
or
x^2 + 7x - 30
= (x-2)(x+9 - 12/(x-2) )
so the sides could be x-2 and x+9 - 12/(x-2)
or
x^2 + 7x - 30
= (x+5)((x+2 - 40/(x+5) )
so the sides could be x+5 and x+2 - 40/(x+5)
or ...
as you can see there is no unique solution, I could pick any binomial for a first side, do a division and get the 2nd side.
Illustrate with numbers
Suppose you have an area of 30
so the sides could be 5 and 6
or
the sides could be 3 and 10
or
the sides could be 1 and 30
or
4.5 and 6 2/3
etc, can you see my point?