To find the dimensions (length and width) of the swimming pool using the given expression for its area, we first need to simplify the area equation. The expression you provided seems to have a small typo, but based on what you wrote, we'll consider it as a quadratic expression. We will assume the expression for the area is:
\[ A = 3x^2 + 10x - 83 \]
To find the dimensions, we can factor this quadratic expression. We need to express this in a form that allows us to find the dimensions as the product of two binomials.
Let's first factor \( A = 3x^2 + 10x - 83 \):
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Multiply the coefficient of \( x^2 \) (which is 3) by the constant term (which is -83): \[ 3 \times -83 = -249 \]
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We need to find two numbers that multiply to -249 and add to 10 (the coefficient of \( x \)). The pair of numbers that meet this requirement are \( 19 \) and \( -13 \) (since \( 19 \times -13 = -247 \) and \( 19 + (-13) = 6 \)).
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We can rewrite the middle term using these numbers: \[ 3x^2 + 19x - 13x - 83 \]
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Next, we will factor by grouping: \[ (3x^2 + 19x) + (-13x - 83) = x(3x + 19) - 13(3x + 19) \] \[ = (3x + 19)(x - 13) \]
So the expression can be factored into: \[ A = (3x + 19)(x - 13) \]
Now, we can interpret the factors \( 3x + 19 \) and \( x - 13 \) as the dimensions of the pool.
Thus, the possible dimensions (length and width) of the swimming pool are:
- \( 3x + 19 \)
- \( x - 13 \)
If you have a specific value for \( x \), you can substitute it into these expressions to find the actual measurements of the swimming pool. If there is a specific context (like constraints or conditions) for \( x \), please provide those for more precise answers.