Calculate Rectangle Area: Solve for X Given X-6 Dimensions

Q: How do I know to use (X-3)² instead of (X-6)²?

When completing the square for $ X^2 - 6X $, take half of the coefficient of the X term. Since the coefficient is -6, half of that is -3, giving us $ (X-3)^2 $.

Q: Where does the -9 come from in the final answer?

When you expand $ (X-3)^2 $, you get $ X^2 - 6X + 9 $. But we only want $ X^2 - 6X $, so we must subtract the extra 9: $ (X-3)^2 - 9 $.

Q: Can I solve this without completing the square?

Yes! You could leave the answer as $ s = X^2 - 6X $, but the question specifically asks you to choose from completed square forms. Always match the format the question requires.

Quadratic Expressions with Completing the Square

Given a rectangle whose side is smaller by 6 than the other side. We mark the area of the rectangle with S

and the large side with X

Check the correct argument:

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the correct expression for the rectangle's area

00:03 We'll use the formula for calculating rectangle area (side times side)

00:12 Open parentheses properly, multiply by each factor

00:19 Add 9 to each side of the equation

00:33 Use the shortened multiplication formulas to convert parentheses

00:36 Isolate the area back

00:40 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

Given a rectangle whose side is smaller by 6 than the other side. We mark the area of the rectangle with S

and the large side with X

Check the correct argument:

Step-by-step solution

To solve this problem, we need to express the area of the rectangle in terms of $X$ , accounting for the relationship between its sides.

Step 1: Calculate the area of the rectangle.
The area $s$ of a rectangle is obtained by multiplying the length by the width:

$s = X \times (X - 6)$

Expanding the expression gives us:

$s = X^2 - 6X$

We want to express this in a form that matches the given answer choices. Recognizing the square of a binomial will help us reformulate:

$s = X^2 - 6X$ can be related to a square of a binomial by adjusting it:

Step 2: Recast into a recognizable square:
We want to find a relation to $(x-3)^2$ , so bear in mind:

$(X - 3)^2 = X^2 - 6X + 9$

Therefore, rearranging in the form of $(X - 3)^2$ , we derive:

$s = (X - 3)^2 - 9$

Therefore, the area of the rectangle, expressed in a way to match the correct choices, is $s = (x - 3)^2 - 9$ , which corresponds to choice 3 from the given options.

The correct choice is: Choice 3: $s=(x-3)^2-9$ .

Final Answer

$s=(x-3)^2-9$

Key Points to Remember

Essential concepts to master this topic

Area Formula: Rectangle area equals length times width: $s = X \times (X-6)$
Technique: Expand to $s = X^2 - 6X$ , then complete square using half coefficient
Check: Verify $(X-3)^2 - 9 = X^2 - 6X$ expands correctly ✓

Common Mistakes

Avoid these frequent errors

Forgetting the constant term when completing the square
Don't write $X^2 - 6X = (X-3)^2$ directly = missing the 9! When you complete the square, $(X-3)^2 = X^2 - 6X + 9$ , so you need to subtract 9. Always remember: $X^2 - 6X = (X-3)^2 - 9$ .

Practice Quiz

Test your knowledge with interactive questions

Declares the given expression as a sum

$$ (7b-3x)^2 $$

FAQ

Everything you need to know about this question

Why do we need to complete the square for this rectangle problem?

The question asks you to match one of the given answer choices, which are all in completed square form. Starting with $s = X^2 - 6X$ , we rewrite it to match the format $(X-3)^2 - 9$ .

How do I know to use (X-3)² instead of (X-6)²?

When completing the square for $X^2 - 6X$ , take half of the coefficient of the X term. Since the coefficient is -6, half of that is -3, giving us $(X-3)^2$ .

Where does the -9 come from in the final answer?

When you expand $(X-3)^2$ , you get $X^2 - 6X + 9$ . But we only want $X^2 - 6X$ , so we must subtract the extra 9: $(X-3)^2 - 9$ .

Can I solve this without completing the square?

Yes! You could leave the answer as $s = X^2 - 6X$ , but the question specifically asks you to choose from completed square forms. Always match the format the question requires.

How do I check if my completed square is correct?

Expand your completed square form back to standard form. For example: $(X-3)^2 - 9 = X^2 - 6X + 9 - 9 = X^2 - 6X$ . This should match your original expression!

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