Solve for X: Area vs. Perimeter in Two Altered Square Equations

Q: Why can't the answer be negative 4?

Great observation! While $ x^2 = 16 $ gives us both x = 4 and x = -4, a side length must be positive. You can't have a square with negative side length!

Q: How do I know which square is larger?

The problem states "one side of the squares is larger by 2 than the other." This means if the small square has side x, the large square has side x + 2.

Q: What's the difference between area and perimeter formulas?

For a square with side length s:Area = s × s = $ s^2 $Perimeter = s + s + s + s = $ 4s $Area is square units, perimeter is linear units.

Quadratic Equations with Area-Perimeter Relations

Given two squares, one side of the squares is larger by 2 than the other. The area of the large square is larger than the perimeter of the small square by 20

Find the length of the small square

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:13 Let's find the value of X. Are you ready?

00:17 First, we use the formula for the area of a square, which is side times side.

00:24 Next, let's expand those brackets using the distributive property. Take it step by step.

00:33 Remember, the perimeter of a square is the sum of all its sides.

00:38 In a square, all sides are equal. Keep that in mind!

00:43 Now, substitute the values into the equation and try to solve for X.

00:51 Let's simplify what we can. You're doing great!

00:57 Isolate X by rearranging the equation. Almost there!

01:02 And there you have it! That's the solution to the problem.

01:06 Remember, X is positive because it represents a length of a side.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given two squares, one side of the squares is larger by 2 than the other. The area of the large square is larger than the perimeter of the small square by 20

Find the length of the small square

Step-by-step solution

To find the length of the smaller square, we need to solve the equation derived from the problem statement:

Step 1: Write down the equation for the area of the larger square: $(x + 2)^2$ .
Step 2: Write down the equation for the perimeter of the smaller square: $4x$ .
Step 3: Set up the equation as given: $(x + 2)^2 = 4x + 20$ .

Let's solve the equation:

Step 1: Expand $(x + 2)^2$ :

(x + 2)^2 = x^2 + 4x + 4

Step 2: Rewrite the equation substituting the expanded form:

x^2 + 4x + 4 = 4x + 20

Step 3: Simplify by eliminating $4x$ from both sides:

x^2 + 4 = 20

Step 4: Subtract 4 from both sides:

x^2 = 16

Step 5: Take the square root of both sides:

x = 4

x = -4

Since $x$ must be positive, we have:

x = 4

Thus, the length of the side of the smaller square is $4$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Setup: Area of large square equals perimeter of small square plus 20
Technique: Expand $(x+2)^2 = x^2 + 4x + 4$ before solving
Check: Verify x = 4: Area = 36, Perimeter = 16, difference = 20 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to expand the squared binomial
Don't try to solve $(x+2)^2 = 4x + 20$ without expanding first = stuck with unsolvable form! The squared term hides the quadratic nature. Always expand $(x+2)^2$ to $x^2 + 4x + 4$ before proceeding.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that has the same value as the following:

$$ (x+3)^2 $$

FAQ

Everything you need to know about this question

Why can't the answer be negative 4?

Great observation! While $x^2 = 16$ gives us both x = 4 and x = -4, a side length must be positive. You can't have a square with negative side length!

How do I know which square is larger?

The problem states "one side of the squares is larger by 2 than the other." This means if the small square has side x, the large square has side x + 2.

What's the difference between area and perimeter formulas?

For a square with side length s:

Area = s × s = $s^2$
Perimeter = s + s + s + s = $4s$

Area is square units, perimeter is linear units.

Why does the area equal perimeter plus 20?

This is just the given condition in the problem! The large square's area is 20 units more than the small square's perimeter. We use this relationship to set up our equation.

Can I solve this without expanding the binomial?

It's very difficult! While you could try taking the square root of both sides, you'd get $x + 2 = \sqrt{4x + 20}$ , which is much harder to solve than expanding first.

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