Solve for X: Calculating the Area of a Rectangle with (x-7)(x+7)

Q: What is the difference of squares pattern?

The difference of squares follows the pattern $ (a-b)(a+b) = a^2 - b^2 $. In our case, $ (x-7)(x+7) = x^2 - 7^2 = x^2 - 49 $.

Q: How do I know which dimensions go with which sides?

From the diagram, we can see the rectangle has dimensions 5 and 19. The total area is $ 5 \times 19 = 95 $, which must equal $ (x-7)(x+7) $.

Q: Can x be negative in a geometry problem?

Mathematically, both solutions are valid for the equation $ x^2 - 49 = 95 $. However, if x represents a physical measurement like length, only positive values make sense in real-world contexts.

Q: What if I expanded (x-7)(x+7) incorrectly?

Remember: $ (x-7)(x+7) = x^2 + 7x - 7x - 49 = x^2 - 49 $. The middle terms cancel out because they're opposites: +7x and -7x.

Quadratic Equations with Difference of Squares

The area of the rectangle below is equal to $(x-7)(x+7)$ .

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 We'll use the formula for calculating rectangle area (side times side)

00:10 We'll substitute appropriate values according to the given data and solve for the area

00:13 This is the rectangle's area

00:17 We'll substitute the rectangle's area in the equation and solve for X

00:28 We'll use the square of binomials formula to expand the brackets

00:47 Calculate 7 squared

00:56 Isolate X

01:09 Take the square root to find possible solutions for X

01:14 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The area of the rectangle below is equal to $(x-7)(x+7)$ .

Calculate x.

Step-by-step solution

To find $x$ , we'll begin by calculating the area of the rectangle using its dimensions:

Step 1: Calculate the area $A$ using the dimensions 5 and 19:
$A = 5 \times 19 = 95$
Step 2: Set up the equation using the difference of squares:
$(x-7)(x+7) = x^2 - 49$
Step 3: Equate to the area calculated:
$x^2 - 49 = 95$
Step 4: Solve for $x^2$ :
$x^2 = 95 + 49 = 144$
Step 5: Solve for $x$ by taking the square root:
$x = \pm \sqrt{144} = \pm 12$

Therefore, the solutions for $x$ are $x = \pm 12$ .

Final Answer

$\pm12$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: $(x-7)(x+7) = x^2 - 49$ is difference of squares
Area Method: Calculate rectangle area: $5 \times 19 = 95$
Verification: Check both solutions: $(12-7)(12+7) = 5 \times 19 = 95$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to take both positive and negative square roots
Don't solve $x^2 = 144$ and only write $x = 12$ ! This misses half the solution because $(-12)^2 = 144$ too. Always remember $x = \pm\sqrt{144} = \pm 12$ when solving quadratic equations.

Practice Quiz

Test your knowledge with interactive questions

Solve:

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

FAQ

Everything you need to know about this question

Why does this problem have two answers?

When you square a number, both positive and negative values give the same result! Since 12² = 144 and (-12)² = 144, both $x = 12$ and $x = -12$ satisfy our equation.

What is the difference of squares pattern?

The difference of squares follows the pattern $(a-b)(a+b) = a^2 - b^2$ . In our case, $(x-7)(x+7) = x^2 - 7^2 = x^2 - 49$ .

How do I know which dimensions go with which sides?

From the diagram, we can see the rectangle has dimensions 5 and 19. The total area is $5 \times 19 = 95$ , which must equal $(x-7)(x+7)$ .

Can x be negative in a geometry problem?

Mathematically, both solutions are valid for the equation $x^2 - 49 = 95$ . However, if x represents a physical measurement like length, only positive values make sense in real-world contexts.

What if I expanded (x-7)(x+7) incorrectly?

Remember: $(x-7)(x+7) = x^2 + 7x - 7x - 49 = x^2 - 49$ . The middle terms cancel out because they're opposites: +7x and -7x.

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