Solve for X: Understanding the Rectangle's Area via (4x+9)(4x-9)

Q: How do I recognize when to use difference of squares?

Look for the pattern (something + number)(same thing - same number). In this case, $ (4x+9)(4x-9) $ has the same first term (4x) and the same second term (9) with opposite signs.

Q: Why do we get both positive and negative answers?

When we solve $ x^2 = 9 $, we take the square root of both sides. Since both 3² and (-3)² equal 9, we get two valid solutions: x = ±3.

Q: How can a rectangle have a negative dimension?

In pure geometry, lengths can't be negative. But in algebraic contexts like this problem, we accept both solutions because the mathematical relationship $ (4x+9)(4x-9) = 63 $ works for both x = 3 and x = -3.

Q: What if I can't see the difference of squares pattern?

Use FOIL as a backup! Expand $ (4x+9)(4x-9) $ step by step: First + Outer + Inner + Last. You'll get $ 16x^2 + 36x - 36x - 81 $, and notice the middle terms cancel out.

Q: How do I check if x = -3 really works?

Substitute back: $ (4(-3)+9)(4(-3)-9) = (-12+9)(-12-9) = (-3)(-21) = 63 $. It matches the rectangle area of 9 × 7 = 63 ✓

Difference of Squares with Rectangle Area

The area of a rectangle is equal to $(4x+9)(4x-9)$ .

Calculate x.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Use the formula for calculating rectangle area (side times side)

00:07 Substitute the side values according to the given data, and solve for the area

00:16 Substitute the area value according to the given data and solve for X

00:25 Use the shortened multiplication formulas to expand the brackets

00:41 Calculate 4 squared and 9 squared

00:46 Isolate X

01:19 Take the square root to find possible solutions for X

01:24 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The area of a rectangle is equal to $(4x+9)(4x-9)$ .

Calculate x.

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Simplify the expression $(4x+9)(4x-9)$ using the difference of squares formula.
Step 2: Calculate the area using the rectangle's given dimensions, 9 and 7.
Step 3: Equate the expressions and solve for $x$ .

Step 1: Simplifying $(4x+9)(4x-9)$ .
This expression is a difference of squares, so it simplifies as follows:

$(4x+9)(4x-9) = (4x)^2 - 9^2 = 16x^2 - 81.$

Step 2: Calculate the area using given rectangle dimensions: 9 and 7.
The area $A$ of the rectangle is:

$A = 9 \times 7 = 63.$

Step 3: Equate the two expressions:
$16x^2 - 81 = 63.$
To solve for $x$ , start by adding 81 to both sides:

$16x^2 = 144.$

Divide both sides by 16 to isolate $x^2$ :

$x^2 = 9.$

Take the square root of both sides to solve for $x$ :

$x = \pm\sqrt{9} = \pm3.$

Therefore, the solution to the problem is $x = \pm3$ .

Final Answer

$\operatorname{\pm}3$

Key Points to Remember

Essential concepts to master this topic

Formula: Difference of squares: $(a+b)(a-b) = a^2 - b^2$
Technique: $(4x+9)(4x-9) = 16x^2 - 81$ using the pattern
Check: Substitute x = 3: area = $16(9) - 81 = 63$ matches 9×7 ✓

Common Mistakes

Avoid these frequent errors

Expanding (4x+9)(4x-9) using FOIL method
Don't use FOIL to get $16x^2 + 36x - 36x - 81$ = extra unnecessary work! This takes much longer and creates chances for arithmetic errors. Always recognize the difference of squares pattern and use $(a+b)(a-b) = a^2 - b^2$ directly.

Practice Quiz

Test your knowledge with interactive questions

Solve:

$$ (2+x)(2-x)=0 $$

FAQ

Everything you need to know about this question

How do I recognize when to use difference of squares?

Look for the pattern (something + number)(same thing - same number). In this case, $(4x+9)(4x-9)$ has the same first term (4x) and the same second term (9) with opposite signs.

Why do we get both positive and negative answers?

When we solve $x^2 = 9$ , we take the square root of both sides. Since both 3² and (-3)² equal 9, we get two valid solutions: x = ±3.

How can a rectangle have a negative dimension?

In pure geometry, lengths can't be negative. But in algebraic contexts like this problem, we accept both solutions because the mathematical relationship $(4x+9)(4x-9) = 63$ works for both x = 3 and x = -3.

What if I can't see the difference of squares pattern?

Use FOIL as a backup! Expand $(4x+9)(4x-9)$ step by step: First + Outer + Inner + Last. You'll get $16x^2 + 36x - 36x - 81$ , and notice the middle terms cancel out.

How do I check if x = -3 really works?

Substitute back: $(4(-3)+9)(4(-3)-9) = (-12+9)(-12-9) = (-3)(-21) = 63$ . It matches the rectangle area of 9 × 7 = 63 ✓

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