3. The rectangle has an area of 1 square unit. What is the value of x?
The length of the rectangle is (3)/(x+2)
The width is (2)/(x-3)
I can't seem to figure this out, could some please help me? I'm in urgent need of help as this assignment is due tomorrow...
Thanks for any help!
well, since area = width * length,
3/(x+2) * 2/(x-3) = 1
6 = (x+2)(x-3)
x^2 - x - 12 = 0
(x-4)(x+3) = 0
x = 4 or -3
-3 is no good, since we need positive values for the sides, so x=4.
Check:
3/6 * 2/1 = 1
To solve for the value of x, we need to set up an equation using the given information.
The formula for the area of a rectangle is A = length * width.
Given:
Area = 1 square unit
Length = 3/(x+2)
Width = 2/(x-3)
Plugging in the given values, we get:
1 = (3/(x+2)) * (2/(x-3))
Now let's solve this equation step by step:
Step 1: Multiply both sides of the equation by (x+2)(x-3) to eliminate the denominators:
1 * (x+2)(x-3) = (3/(x+2)) * (2/(x-3)) * (x+2)(x-3)
Simplifying, we have:
(x+2)(x-3) = 3 * 2
Step 2: Expand the left side of the equation:
x^2 - 3x + 2x - 6 = 6
Simplifying further:
x^2 - x - 6 = 6
Step 3: Move the constant term to the right side of the equation:
x^2 - x - 6 - 6 = 0
Simplifying:
x^2 - x - 12 = 0
Step 4: Factor the quadratic equation:
(x - 4)(x + 3) = 0
Step 5: Set each factor equal to zero and solve for x:
x - 4 = 0 or x + 3 = 0
Solving these two equations, we find:
x = 4 or x = -3
So the possible values of x are 4 and -3.
To find the value of x, we can start by setting up the equation for the area of the rectangle.
The area of a rectangle is given by the formula: Area = length * width
In this case, we have:
Area = 1 square unit
Length of the rectangle = 3 / (x+2)
Width of the rectangle = 2 / (x-3)
Now, let's substitute these values into the area formula:
1 = (3 / (x+2)) * (2 / (x-3))
Next, let's simplify the equation by cross-multiplying and getting rid of the fractions:
1 = (6 / ((x+2) * (x-3)))
Now, let's multiply both sides of the equation by ((x+2) * (x-3)) to isolate the variable:
((x+2) * (x-3)) = 6
Let's expand the left side of the equation:
(x^2 - x - 6) = 6
Rearranging the equation:
x^2 - x - 12 = 0
Now, we have a quadratic equation. To solve for x, we can use factoring, completing the square, or the quadratic formula. In this case, we can factor the quadratic equation:
(x - 4)(x + 3) = 0
Setting each factor equal to zero:
x - 4 = 0 or x + 3 = 0
Solving for x in each equation:
x = 4 or x = -3
Therefore, the value of x can either be 4 or -3.
You can double-check your answer by substituting both values back into the original equations for length and width to ensure that the area of the rectangle is indeed 1 square unit.