Check the True Statement: Geometry Challenge with S = X² and P = 4X

Q: Why do we add 4 to P + S in the correct answer?

Adding 4 creates a perfect square trinomial! Since $ P + S = 4X + X^2 $, adding 4 gives us $ X^2 + 4X + 4 $, which factors perfectly as $ (X + 2)^2 $.

Q: How do I remember the difference between area and perimeter?

Area measures the space inside the square, so we multiply length × width = X². Perimeter measures the border around the square, so we add all four sides = 4X.

Q: What happens if I expand (X + 4)² instead?

Expanding $ (X + 4)^2 $ gives $ X^2 + 8X + 16 $. This doesn't match $ P + S + 4 = X^2 + 4X + 4 $ because the middle term is 8X instead of 4X.

Q: Can I solve this by plugging in a specific value for X?

Yes! Try X = 3: P = 12, S = 9, so P + S + 4 = 25. Check: $ (3 + 2)^2 = 5^2 = 25 $ ✓. This confirms the correct answer!

Q: Why is this called a perfect square trinomial?

A perfect square trinomial has the pattern $ a^2 + 2ab + b^2 = (a + b)^2 $. Here, $ X^2 + 4X + 4 $ fits this pattern with a = X and b = 2.

Square Formulas with Algebraic Verification

Given a square of side length X

We will mark the area of the square by S and the perimeter of the square by P

Check the correct statement

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's identify the correct statement.

00:12 Start by calculating the square's area. Use the side times itself.

00:17 Now, find the square's perimeter. Multiply the side length by four.

00:25 Next, add the perimeter and the area together.

00:30 Then, add four to your result.

00:42 Break down four into two times two.

00:48 Use quick multiplication tricks to find the terms in brackets.

00:53 And that's how you solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given a square of side length X

We will mark the area of the square by S and the perimeter of the square by P

Check the correct statement

Step-by-step solution

To solve this problem, we begin by calculating the area and the perimeter of the square:

Perimeter, $P = 4X$
Area, $S = X^2$

The sum of $P$ and $S$ is:

$P + S = 4X + X^2$

We need to evaluate the choice that correctly equates:

Given the choice $P + S + 4 = (X + 2)^2$ , we expand and simplify:

$(X + 2)^2 = X^2 + 4X + 4$
This expression matches $P + S + 4 = X^2 + 4X + 4$ .

Thus, the expression $P + S + 4 = (X + 2)^2$ is correct.

Therefore, the solution to the problem is $P+S+4=(x+2)^2$ .

Final Answer

$P+S+4=(x+2)^2$

Key Points to Remember

Essential concepts to master this topic

Formulas: Square area S = X² and perimeter P = 4X
Technique: Expand (X + 2)² = X² + 4X + 4 to match expressions
Check: Substitute values to verify P + S + 4 equals the expanded form ✓

Common Mistakes

Avoid these frequent errors

Confusing area and perimeter formulas
Don't mix up S = 4X and P = X² = wrong calculations! This switches the basic square formulas and leads to incorrect algebraic expressions. Always remember area uses the exponent (S = X²) while perimeter uses multiplication (P = 4X).

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 do we add 4 to P + S in the correct answer?

Adding 4 creates a perfect square trinomial! Since $P + S = 4X + X^2$ , adding 4 gives us $X^2 + 4X + 4$ , which factors perfectly as $(X + 2)^2$ .

How do I remember the difference between area and perimeter?

Area measures the space inside the square, so we multiply length × width = X². Perimeter measures the border around the square, so we add all four sides = 4X.

What happens if I expand (X + 4)² instead?

Expanding $(X + 4)^2$ gives $X^2 + 8X + 16$ . This doesn't match $P + S + 4 = X^2 + 4X + 4$ because the middle term is 8X instead of 4X.

Can I solve this by plugging in a specific value for X?

Yes! Try X = 3: P = 12, S = 9, so P + S + 4 = 25. Check: $(3 + 2)^2 = 5^2 = 25$ ✓. This confirms the correct answer!

Why is this called a perfect square trinomial?

A perfect square trinomial has the pattern $a^2 + 2ab + b^2 = (a + b)^2$ . Here, $X^2 + 4X + 4$ fits this pattern with a = X and b = 2.

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