the area of a rectangular pool table is 6x^2 - 11x+3. Factor 6x^2 - 11x+3 to determine the dimensions (length and width) of the rectangular pool.
The factored form of 6x^2 - 11x+3 is (3x-1)(2x-3).
Therefore, the length of the rectangular pool is 3x-1 and the width is 2x-3.
To factor the quadratic expression 6x^2 - 11x + 3, we need to find two binomials that, when multiplied together, give us this quadratic expression.
First, we look for factors of the coefficient of x^2 (which is 6). The factors of 6 are 1, 2, 3, and 6.
Next, we look for factors of the constant term (which is 3). The factors of 3 are 1 and 3.
Now, we need to find two numbers that multiply to give us 6 and add up to -11 (the coefficient of x). The numbers that satisfy this condition are -9 and -2.
So, we can write the quadratic expression as the product of two binomials:
6x^2 - 11x + 3 = (2x - 1)(3x - 3)
The dimensions of the rectangular pool table are given by the coefficients in these binomials. Therefore, the length is 3x - 3, and the width is 2x - 1.
To factor the quadratic expression 6x^2 - 11x + 3, we need to find two binomials whose product will give us this quadratic expression.
We can start by looking for two binomials in the form (ax + b)(cx + d), where a, b, c, and d are integers.
First, let's multiply the first terms of each binomial: (ax)(cx) = acx^2. In this case, ac = 6, so we have narrowed it down to the following possibilities:
(2x + ?)(3x + ?)
(1x + ?)(6x + ?)
Next, we'll look at the last terms of each binomial: (b)(d) = 3. We need to find two integers, b and d, whose product is 3. The possibilities are:
1 × 3
-1 × -3
Now, we need to find the middle term of the quadratic expression (-11x) by adding or subtracting the outer and inner products of the binomials.
Option 1: (2x + ?)(3x + ?)
Possible combinations for the outer and inner products:
2x × ? + 3x × ?
6x^2 + 2x × ? + 3x × ? + ?
We can see that the only way to get -11x is by multiplying 2x by -3 and 3x by 1, resulting in (-6x + 3x) = -3x.
So, our first binomial is (2x - 1), and the second binomial is (3x + 1).
Option 2: (1x + ?)(6x + ?)
Possible combinations for the outer and inner products:
1x × ? + 6x × ?
6x^2 + 1x × ? + 6x × ? + ?
From this, it is clear that we cannot achieve -11x as the middle term, so option 2 is not correct.
Therefore, we can factor the quadratic expression as: (2x - 1)(3x + 1).
Since this represents the dimensions of a rectangular pool, the length is (2x - 1) and the width is (3x + 1).