Question
the area of a rectangular carpet is given by the trinomial 5x^2-3x-14. what are the possible dimensions of the carpet? use factoring
Answers
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GPT 3.5
To find the dimensions of the carpet, we need to factor the trinomial 5x^2 - 3x - 14.
We can factor this trinomial by looking for two binomials that multiply to give us 5x^2 - 3x - 14. The binomials will have the form (mx + a)(nx + b).
First, we look for integers m, n, a, and b such that (m)(n) = 5, (a)(b) = -14, and (m)(b) + (n)(a) = -3.
The factors of 5 are 1 and 5.
The factors of -14 are -1 and 14, -2 and 7, or -7 and 2.
By trying different combinations, we find that when m = 5, n = -2, a = -7, and b = 2, we have (5x - 7)(x + 2) = 5x^2 - 3x - 14.
Now that we have factored the trinomial, we can find the possible dimensions of the carpet by setting each binomial equal to zero and solving for x.
Setting 5x - 7 = 0, we find x = 7/5.
Setting x + 2 = 0, we find x = -2.
Therefore, the possible dimensions of the carpet are 7/5 units and -2 units. However, since dimensions cannot be negative, the only possible dimension is 7/5 units.
We can factor this trinomial by looking for two binomials that multiply to give us 5x^2 - 3x - 14. The binomials will have the form (mx + a)(nx + b).
First, we look for integers m, n, a, and b such that (m)(n) = 5, (a)(b) = -14, and (m)(b) + (n)(a) = -3.
The factors of 5 are 1 and 5.
The factors of -14 are -1 and 14, -2 and 7, or -7 and 2.
By trying different combinations, we find that when m = 5, n = -2, a = -7, and b = 2, we have (5x - 7)(x + 2) = 5x^2 - 3x - 14.
Now that we have factored the trinomial, we can find the possible dimensions of the carpet by setting each binomial equal to zero and solving for x.
Setting 5x - 7 = 0, we find x = 7/5.
Setting x + 2 = 0, we find x = -2.
Therefore, the possible dimensions of the carpet are 7/5 units and -2 units. However, since dimensions cannot be negative, the only possible dimension is 7/5 units.