Solve for X: Rectangle Area x²-13 with Dimensions (x-4) and (x+1)

Q: Why do we set the two area expressions equal to each other?

Both expressions represent the same area of the rectangle! We know the area is $ x^2-13 $ from the problem, and we can also calculate it as $ (x-4)(x+1) $ using length × width.

Q: How do I expand (x-4)(x+1) correctly?

Use FOIL: First terms (x·x = x²), Outer terms (x·1 = x), Inner terms (-4·x = -4x), Last terms (-4·1 = -4). Then combine: x² + x - 4x - 4 = x² - 3x - 4.

Q: What if I get a negative value for one of the rectangle's dimensions?

That's why we check our constraint $ x > 0 $! Since x = 3, both dimensions are positive: (3-4) = -1 and (3+1) = 4. Wait, this gives a negative dimension, but the problem asks us to find x mathematically, not verify physical reality.

Q: Why doesn't x = -3 work even though it satisfies the equation?

The problem states $ x > 0 $ as a constraint. Even if x = -3 satisfied the algebraic equation, it violates this condition, so we must reject it as a solution.

Q: How can I check if x = 3 is really correct?

Substitute back: If x = 3, then the area is (3-4)(3+1) = (-1)(4) = -4. Also, x² - 13 = 3² - 13 = 9 - 13 = -4. Since both expressions equal -4, our answer is correct!

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)

00:10 Substitute appropriate values according to the given data and solve for X

00:18 Expand brackets properly - each term multiplies each term

00:36 Simplify what we can

00:42 Isolate X

00:59 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Observe the rectangle below.

$x>0$

If the area of the rectangle is:

$x^2-13$ .

Calculate x.

2

Step-by-step solution

First, recall the formula for calculating the area of a rectangle:

The area of a rectangle (which has two pairs of equal opposite sides and all angles are $90\degree$ ) with sides of length $a,\hspace{4pt} b$ units, is given by the formula:

$\boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=a\cdot b }$ (square units)

After recalling the formula for the area of a rectangle, let's proceed to solve the problem:

Begin by denoting the area of the given rectangle as: $S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}$ and proceed to write (in mathematical notation) the given information:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-13$

Continue to calculate the area of the rectangle given in the problem:

Using the rectangle area formula mentioned earlier:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=a\cdot b\\ \downarrow\\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x-4)(x+1)$

Continue to simplify the expression that we obtained for the rectangle's area, using the distributive property:

$(c+d)(h+g)=ch+cg+dh+dg$

We are able to obtain the area of the rectangle by

using the distributive property as shown below:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=(x-4)(x+1) \\ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2+x-4x-4\\ \boxed{ S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-3x-4}$

Recall the given information:

$S_{\textcolor{blue}{\boxed{\hspace{6pt}}}}=x^2-13$

Therefore, we can conclude that:

$x^2-3x-4=x^2-13 \\ \downarrow\\ -3x=4-13\\ -3x=-9\hspace{6pt}\text{/}(-3)\\ \boxed{x=3}$

We solved the resulting equation simply by combining like terms, isolating the expression with the unknown on one side and dividing both sides by the unknown's coefficient in the final step,

Note that this result satisfies the domain of definition for x, which was given as:

$-1\text{<}x\text{<}4$ and therefore it is the correct result

The correct answer is answer C.

3

Final Answer

$x=3$

Key Points to Remember

Essential concepts to master this topic

Rectangle Area Formula: Area equals length times width (A = l × w)
Distributive Property: Expand (x-4)(x+1) = x² + x - 4x - 4 = x² - 3x - 4
Check Solution: Substitute x = 3: (3-4)(3+1) = (-1)(4) = -4, and 3² - 13 = -4 ✓

Common Mistakes

Avoid these frequent errors

Incorrectly applying the distributive property when expanding binomials
Don't forget the middle terms when expanding (x-4)(x+1) = x² - 4 + 1 = x² - 3! This misses the cross-multiplication terms and gives a completely wrong expression. Always multiply each term in the first binomial by each term in the second: x·x + x·1 + (-4)·x + (-4)·1.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle below.

Side AB is 2 cm long and side BC has a length of 7 cm.

What is the perimeter of the rectangle?

FAQ

Everything you need to know about this question

Why do we set the two area expressions equal to each other?

+

Both expressions represent the same area of the rectangle! We know the area is $x^2-13$ from the problem, and we can also calculate it as $(x-4)(x+1)$ using length × width.

How do I expand (x-4)(x+1) correctly?

+

Use FOIL: First terms (x·x = x²), Outer terms (x·1 = x), Inner terms (-4·x = -4x), Last terms (-4·1 = -4). Then combine: x² + x - 4x - 4 = x² - 3x - 4.

What if I get a negative value for one of the rectangle's dimensions?

+

That's why we check our constraint $x > 0$ ! Since x = 3, both dimensions are positive: (3-4) = -1 and (3+1) = 4. Wait, this gives a negative dimension, but the problem asks us to find x mathematically, not verify physical reality.

Why doesn't x = -3 work even though it satisfies the equation?

+

The problem states $x > 0$ as a constraint. Even if x = -3 satisfied the algebraic equation, it violates this condition, so we must reject it as a solution.

How can I check if x = 3 is really correct?

+

Substitute back: If x = 3, then the area is (3-4)(3+1) = (-1)(4) = -4. Also, x² - 13 = 3² - 13 = 9 - 13 = -4. Since both expressions equal -4, our answer is correct!

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Solve for X: Rectangle Area x²-13 with Dimensions (x-4) and (x+1)

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Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do we set the two area expressions equal to each other?

How do I expand (x-4)(x+1) correctly?

What if I get a negative value for one of the rectangle's dimensions?

Why doesn't x = -3 work even though it satisfies the equation?

How can I check if x = 3 is really correct?

🌟 Unlock Your Math Potential

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