Calculate Rectangle Area with Dimensions (x-2) and (x+4): Algebraic Geometry

Q: Why do we multiply the dimensions instead of adding them?

Area measures the space inside the rectangle. Think of it as counting unit squares - you need length × width squares to fill it completely. Adding gives you perimeter (distance around the outside).

Q: How do I expand (x-2)(x+4) correctly?

Use the FOIL method:First: x × x = x²Outer: x × 4 = 4xInner: -2 × x = -2xLast: -2 × 4 = -8Then combine: $ x^2 + 4x - 2x - 8 = x^2 + 2x - 8 $

Q: What if I get a different sign pattern?

Check your signs carefully! In $ (x-2)(x+4) $, the -2 multiplies both terms in the second binomial: $ -2 × x = -2x $ and $ -2 × 4 = -8 $.

Q: How can I check if x² + 2x - 8 is correct?

Try substituting a simple value like x = 3:Using dimensions: $ (3-2)(3+4) = 1 × 7 = 7 $Using our answer: $ 3^2 + 2(3) - 8 = 9 + 6 - 8 = 7 $ ✓Both give the same result!

Q: Why is the constant term negative?

When you multiply $ -2 × 4 = -8 $, you get a negative result. This happens because one factor is negative (-2) and the other is positive (4). The negative × positive = negative rule applies.

Polynomial Multiplication with Binomial Expressions

Look at the rectangle in the figure.

What is its area?

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Step-by-step video solution

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00:05 Let's calculate the area of this rectangle.

00:09 We will use the formula for the area of a rectangle: length times width.

00:17 First, let's make sure to open those parentheses. Each number will multiply with the other.

00:42 Now, let's do the math to find the products.

00:55 Next, we group the numbers together.

01:00 And that is how we find the solution to our problem!

Step-by-step written solution

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Understand the problem

Look at the rectangle in the figure.

What is its area?

Step-by-step solution

To find the area of the given rectangle, we will multiply its length and width:

The rectangle has dimensions $x - 2$ and $x + 4$ . The area formula for a rectangle is:

\text{Area} = \text{length} \times \text{width}

Substituting the dimensions, we get:

\text{Area} = (x - 2)(x + 4)

Next, we expand this expression:

(x - 2)(x + 4) = x(x + 4) - 2(x + 4)

The expanded terms are:

= x^2 + 4x - 2x - 8

Combining like terms, we obtain:

= x^2 + 2x - 8

Thus, the area of the rectangle is $x^2 + 2x - 8$ .

In terms of the given choices, the correct choice is: $x^2 + 2x - 8$ .

Final Answer

$x^2+2x-8$

Key Points to Remember

Essential concepts to master this topic

Area Formula: Rectangle area equals length times width
FOIL Method: (x-2)(x+4) = x² + 4x - 2x - 8
Check: Combine like terms: 4x - 2x = 2x gives x² + 2x - 8 ✓

Common Mistakes

Avoid these frequent errors

Adding dimensions instead of multiplying
Don't add (x-2) + (x+4) = 2x + 2! This gives perimeter, not area. Area requires multiplication. Always multiply length × width for rectangle area: (x-2)(x+4) = x² + 2x - 8.

Practice Quiz

Test your knowledge with interactive questions

$$ x^2+6x+9=0 $$

What is the value of X?

FAQ

Everything you need to know about this question

Why do we multiply the dimensions instead of adding them?

Area measures the space inside the rectangle. Think of it as counting unit squares - you need length × width squares to fill it completely. Adding gives you perimeter (distance around the outside).

How do I expand (x-2)(x+4) correctly?

Use the FOIL method:

First: x × x = x²
Outer: x × 4 = 4x
Inner: -2 × x = -2x
Last: -2 × 4 = -8

Then combine: $x^2 + 4x - 2x - 8 = x^2 + 2x - 8$

What if I get a different sign pattern?

Check your signs carefully! In $(x-2)(x+4)$ , the -2 multiplies both terms in the second binomial: $-2 × x = -2x$ and $-2 × 4 = -8$ .

How can I check if x² + 2x - 8 is correct?

Try substituting a simple value like x = 3:

Using dimensions: $(3-2)(3+4) = 1 × 7 = 7$
Using our answer: $3^2 + 2(3) - 8 = 9 + 6 - 8 = 7$ ✓

Both give the same result!

Why is the constant term negative?

When you multiply $-2 × 4 = -8$ , you get a negative result. This happens because one factor is negative (-2) and the other is positive (4). The negative × positive = negative rule applies.

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