Trapezoid Area Problem: Express Area with Height 2X and Parallel Sides (X+14) and (3X+7)

Q: Why do I get a quadratic expression for the area?

Because you're multiplying two algebraic expressions together! When you multiply $ (4X+21) \times X $, you get $ 4X^2 + 21X $. The X² term comes from multiplying X by X terms.

Q: Should I simplify the bases before or after using the formula?

Simplify the bases first! Add $ (X+14) + (3X+7) = 4X+21 $ before plugging into the area formula. This makes the multiplication much easier.

Q: Can I factor my final answer?

Yes! $ 4X^2 + 21X = X(4X + 21) $. Both forms are correct, but the expanded form $ 4X^2 + 21X $ is usually preferred for area problems.

Q: What if I multiply the height by each base separately?

That works too! $ \frac{1}{2}[2X(X+14) + 2X(3X+7)] $ gives the same result, but it's more work. Adding bases first is much simpler.

Q: How can I check if my area expression is right?

Substitute a simple value like X = 1. Your expression should give: $ 4(1)^2 + 21(1) = 25 $. Check this matches $ \frac{1}{2} \times (15+10) \times 2 = 25 $ ✓

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the area of the trapezoid using X

00:03 We'll use the formula for calculating trapezoid area

00:06 (Sum of bases(AB+DC) multiplied by height(H)) divided by 2

00:16 We'll substitute appropriate values according to the given data and solve for the area

00:31 We'll simplify what we can

00:37 We'll group the numbers as one factor and X as another factor

00:41 We'll properly distribute - multiply each factor by X

00:47 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

Express the area of the trapezoid by X

2

Step-by-step solution

To express the area of the trapezoid in terms of $X$ , follow these steps:

Step 1: Identify the given values for the trapezoid's dimensions. The top base $b_1$ is $X + 14$ , the bottom base $b_2$ is $3X + 7$ , and the height $h$ is $2X$ .
Step 2: Use the trapezoid area formula $A = \frac{1}{2} \times (b_1 + b_2) \times h$ .
Step 3: Compute the sum of the bases: $(X + 14) + (3X + 7) = X + 14 + 3X + 7 = 4X + 21$ .
Step 4: Calculate the area using the formula: $A = \frac{1}{2} \times (4X + 21) \times (2X)$ .
Step 5: Simplify: $A = \frac{1}{2} \times (4X + 21) \times 2X = (4X + 21) \times X$ .
Step 6: Simplify further by distributing: $A = 4X^2 + 21X$ .

Thus, the area of the trapezoid expressed in terms of $X$ is $4X^2 + 21X$ .

3

Final Answer

$4x^2+21x$

Key Points to Remember

Essential concepts to master this topic

Formula: Area = $\frac{1}{2} \times (b_1 + b_2) \times h$ for trapezoids
Technique: Add bases first: $(X+14) + (3X+7) = 4X+21$
Check: Distribute carefully: $(4X+21) \times X = 4X^2 + 21X$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply by height after adding bases
Don't calculate just $\frac{1}{2} \times (4X+21) = 2X + 10.5$ = missing the height factor! This gives a linear expression instead of the correct quadratic area. Always multiply the sum of bases by the height before dividing by 2.

Practice Quiz

Test your knowledge with interactive questions

Given the following trapezoid:

Calculate the area of the trapezoid ABCD.

FAQ

Everything you need to know about this question

Why do I get a quadratic expression for the area?

+

Because you're multiplying two algebraic expressions together! When you multiply $(4X+21) \times X$ , you get $4X^2 + 21X$ . The X² term comes from multiplying X by X terms.

Should I simplify the bases before or after using the formula?

+

Simplify the bases first! Add $(X+14) + (3X+7) = 4X+21$ before plugging into the area formula. This makes the multiplication much easier.

Can I factor my final answer?

+

Yes! $4X^2 + 21X = X(4X + 21)$ . Both forms are correct, but the expanded form $4X^2 + 21X$ is usually preferred for area problems.

What if I multiply the height by each base separately?

+

That works too! $\frac{1}{2}[2X(X+14) + 2X(3X+7)]$ gives the same result, but it's more work. Adding bases first is much simpler.

How can I check if my area expression is right?

+

Substitute a simple value like X = 1. Your expression should give: $4(1)^2 + 21(1) = 25$ . Check this matches $\frac{1}{2} \times (15+10) \times 2 = 25$ ✓

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Trapezoid Area Problem: Express Area with Height 2X and Parallel Sides (X+14) and (3X+7)

Trapezoid Area with Algebraic Expressions

❤️ Continue Your Math Journey!

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 I get a quadratic expression for the area?

Should I simplify the bases before or after using the formula?

Can I factor my final answer?

What if I multiply the height by each base separately?

How can I check if my area expression is right?

🌟 Unlock Your Math Potential

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