The area of a rectangular painting is given by the trinomial x^2-5x-36. What are the possible dimensions of the painting? Use factoring.
To find the dimensions of the painting, we need to factor the given trinomial. We can do this by finding two numbers that multiply to -36 and add to -5. These numbers are -9 and +4. Therefore, we can write:
x^2-5x-36 = (x-9)(x+4)
The dimensions of the painting are given by the factors of the trinomial, which are (x-9) and (x+4). These represent the length and width of the painting, respectively. To find the possible dimensions, we need to consider the values of x that make each factor equal to zero. These are:
x-9 = 0 => x = 9
x+4 = 0 => x = -4
Therefore, the possible dimensions of the painting are:
Length = x-9 = 9-9 = 0 or x+4 = -4+4 = 0
Width = x+4 = 9+4 = 13 or x-9 = -4-9 = -13
Since the length cannot be zero, the actual dimensions are:
Length = 9 units
Width = 13 units
Alternatively, if we take the absolute value of x, the possible dimensions are:
Length = |x-9| = |9-9| = 0 or |x+4| = |-4+4| = 0
Width = |x+4| = |9+4| = 13 or |x-9| = |-4-9| = 13
So the actual dimensions are still:
Length = 9 units
Width = 13 units
To find the possible dimensions of the painting, we can factor the trinomial x^2 - 5x - 36.
The factored form of the trinomial is:
(x + 4)(x - 9)
To find the possible dimensions, we set each factor equal to zero:
x + 4 = 0 or x - 9 = 0
Solving each equation, we get:
x = -4 or x = 9
Therefore, the possible dimensions of the painting are -4 and 9.