Create an image of a mathematics concept visualization. Show a rectangle with a subtle grid overlaying it to represent the area in square inches. Beside the rectangle, have arrows pointing to its length and width. Label the length as 'x+3' and the width as 'x-2'. Keep the image free of any additional text.

If the area of a rectangle is 84 square inches.How do i find its length and width? Length is (x+3) width is (x-2).

12 by 7

Well, if the area of the rectangle is 84 square inches, we can set up an equation to solve for the length and width.

Area = Length * Width

So, we have (x+3) * (x-2) = 84

Now, let's solve for x...

x^2 + x - 6 = 84

x^2 + x - 90 = 0

Now, we can either solve this equation by factoring, completing the square, or using the quadratic formula. But hey, let's have some fun with this!

Clown Bot joke time: Why did the math book look so sad? Because it had too many problems!

Okay, back to business. Let's solve it using the quadratic formula:

x = (-1 ± √(1^2 - 4(1)(-90))) / (2(1))
x = (-1 ± √(1 + 360)) / 2
x = (-1 ± √361) / 2
x = (-1 ± 19) / 2

So, we can have two possible values for x: x = 9 or x = -10. But since dimensions can't be negative, the length (x+3) and width (x-2) are 12 inches and 7 inches.

Voila! The length is 12 inches and the width is 7 inches.

To find the length and width of the rectangle, we need to solve the equation for the area of a rectangle.

The area of a rectangle is given by the formula: length × width.

Given that the area is 84 square inches, we can set up the equation:

(x + 3) × (x - 2) = 84

To solve for x, we can expand the equation:

x^2 + x - 6 = 84

Rearranging the equation to get the quadratic equation form:

x^2 + x - 90 = 0

Now we can solve the quadratic equation. This equation can be factored into:

(x - 9)(x + 10) = 0

Setting each factor equal to zero:

x - 9 = 0 or x + 10 = 0

Solving for x in each equation:

x = 9 or x = -10

Since the length and width cannot have negative values, we can reject the negative value. Thus, x = 9.

Using x = 9, we can substitute back into the expressions for the length and width:

Length = x + 3 = 9 + 3 = 12 inches
Width = x - 2 = 9 - 2 = 7 inches

Therefore, the length of the rectangle is 12 inches, and the width is 7 inches.

To find the length and width of the rectangle, you need to use the given information that the area is 84 square inches and that the length is represented by (x+3) and the width is represented by (x-2).

The formula for the area of a rectangle is Length × Width. In this case, the area is given as 84 square inches. So, we can set up the equation:

Length × Width = 84

Substituting the expressions for length and width, the equation becomes:

(x+3) × (x-2) = 84

Now, let's solve this equation to find the value of x:

Using the distributive property: (x+3) × (x-2) = x × x - 2 × x + 3 × x - 3 × 2 = x^2 - 2x + 3x - 6 = x^2 + x - 6

Rearranging the equation: x^2 + x - 6 - 84 = 0
x^2 + x - 90 = 0

Now we have a quadratic equation. We can solve it by factoring or using the quadratic formula.

Factoring the quadratic equation:
(x + 10)(x - 9) = 0

Setting each factor equal to zero:
x + 10 = 0, x - 9 = 0

Solving for x:
x = -10, x = 9

Since the dimensions of a rectangle cannot be negative, we discard the solution x = -10.

Therefore, the value of x is 9.

To find the length and width, substitute the value of x (9) into the expressions (x+3) and (x-2):

Length = 9 + 3 = 12 inches
Width = 9 - 2 = 7 inches

So, the length of the rectangle is 12 inches and the width is 7 inches.

area = length*width

(x+3)(x-2) = 84
x^2 + x - 6 = 84
x^2 + x - 90 = 0
(x+10)(x-9) = 0

x=9, so

length=12
width=7