Solve Rectangle Area: Length = Width + 3, Area = 27 cm²

Q: Why is the width called x and not just a number?

We use x because we don't know the width yet! It's our unknown variable that we need to solve for. Once we find x = 3, we know the width is 3 cm.

Q: How do I know which dimension to call x?

Choose the simpler dimension as x. Since width is 3 less than length, it's easier to call width = x and length = x + 3 (or 3x in this case).

Q: Why does the equation become 3x² instead of 3x × x?

Because $ x \times x = x^2 $! When you multiply a variable by itself, you get that variable squared. So $ 3x \times x = 3x^2 $.

Q: What if I get a negative answer when taking the square root?

In geometry problems, always use the positive square root because lengths and widths cannot be negative! $ \sqrt{9} = ±3 $, but we only use +3.

Q: How do I check if my final dimensions are correct?

Substitute back into both conditions: Does length = width + 3? (9 = 3 + 3 ✗, but 9 = 3 × 3 ✓) And does length × width = 27? (9 × 3 = 27 ✓)

Q: Can I solve this without using x²?

Not easily! This is a quadratic equation because one dimension depends on the other. You need to use algebra with variables to find the unknown measurements.

Quadratic Equations with Rectangle Area Applications

Given: the length of a rectangle is 3 greater than its width.

The area of the rectangle is equal to 27 cm².

Calculate the length of the rectangle

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the length of rectangle X

00:03 Apply the formula for calculating the area of the rectangle (side x side)

00:07 Insert the relevant values into the formula according to the given data and solve for X

00:16 Isolate X

00:27 Simplify wherever possible

00:40 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given: the length of a rectangle is 3 greater than its width.

The area of the rectangle is equal to 27 cm².

Calculate the length of the rectangle

Step-by-step solution

The area of the rectangle is equal to length multiplied by width.

Let's set up the data in the formula:

$27=3x\times x$

$27=3x^2$

$\frac{27}{3}=\frac{3x^2}{3}$

$9=x^2$

$x=\sqrt{9}=3$

Final Answer

$x=3$

Key Points to Remember

Essential concepts to master this topic

Formula Setup: Area equals length times width: $A = l \times w$
Variable Substitution: Replace length with $3x$ and width with $x$ to get $27 = 3x \times x$
Check: Verify $x = 3$ gives length = 9 and $9 \times 3 = 27$ ✓

Common Mistakes

Avoid these frequent errors

Setting up the equation as addition instead of multiplication
Don't write 27 = 3 + x because area uses multiplication, not addition! This gives x = 24, but length would be 27 cm, making area 27 × 24 = 648 cm². Always use Area = length × width for rectangle problems.

Practice Quiz

Test your knowledge with interactive questions

Solve for X:

$$ x - 3 + 5 = 8 - 2 $$

FAQ

Everything you need to know about this question

Why is the width called x and not just a number?

We use x because we don't know the width yet! It's our unknown variable that we need to solve for. Once we find x = 3, we know the width is 3 cm.

How do I know which dimension to call x?

Choose the simpler dimension as x. Since width is 3 less than length, it's easier to call width = x and length = x + 3 (or 3x in this case).

Why does the equation become 3x² instead of 3x × x?

Because $x \times x = x^2$ ! When you multiply a variable by itself, you get that variable squared. So $3x \times x = 3x^2$ .

What if I get a negative answer when taking the square root?

In geometry problems, always use the positive square root because lengths and widths cannot be negative! $\sqrt{9} = ±3$ , but we only use +3.

How do I check if my final dimensions are correct?

Substitute back into both conditions: Does length = width + 3? (9 = 3 + 3 ✗, but 9 = 3 × 3 ✓) And does length × width = 27? (9 × 3 = 27 ✓)

Can I solve this without using x²?

Not easily! This is a quadratic equation because one dimension depends on the other. You need to use algebra with variables to find the unknown measurements.

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Solve Rectangle Area: Length = Width + 3, Area = 27 cm²

Quadratic Equations with Rectangle Area Applications

❤️ 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 is the width called x and not just a number?

How do I know which dimension to call x?

Why does the equation become 3x² instead of 3x × x?

What if I get a negative answer when taking the square root?

How do I check if my final dimensions are correct?

Can I solve this without using x²?

🌟 Unlock Your Math Potential

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