Solve (8+3x)² = (5x+3)² - (4x)²: Perfect Square Equation

Q: When should I use the difference of squares formula?

Use $ a^2 - b^2 = (a-b)(a+b) $ whenever you see two squared terms being subtracted, like $ (5x+3)^2 - (4x)^2 $. This simplifies the problem much faster than expanding!

Q: Why didn't we expand the left side using difference of squares?

The left side $ (8+3x)^2 $ is just one square term, not a difference. The difference of squares formula only works when you have subtraction between two squares.

Q: How do I factor (x+3)(9x+3) correctly?

Use the FOIL method: First terms: $ x \cdot 9x = 9x^2 $, Outer: $ x \cdot 3 = 3x $, Inner: $ 3 \cdot 9x = 27x $, Last: $ 3 \cdot 3 = 9 $. Result: $ 9x^2 + 30x + 9 $

Q: What if I get a mixed number as my answer?

Mixed numbers like $ -3\frac{1}{18} $ are perfectly valid! Convert to check: $ -3\frac{1}{18} = \frac{-55}{18} $. Always verify by substituting back into the original equation.

Q: Can I solve this by taking square roots of both sides?

No! Taking square roots only works when you have $ (expression)^2 = number $. Here we have $ (8+3x)^2 = (5x+3)^2 - (4x)^2 $, so the right side isn't just a number.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:06 Use the shortened multiplication formulas to open the brackets

00:18 When 8 is A

00:21 And 3X is B

00:25 Substitute according to the formula and solve

00:39 Solve the squares and multiplications

01:09 Use the shortened multiplication formulas to open the brackets

01:30 Solve the squares and multiplications

01:58 Collect terms

02:05 Reduce what's possible

02:17 Isolate X

02:51 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

$(8+3x)^2=(5x+3)^2-(4x)^2$

$x=\text{?}$

2

Step-by-step solution

To solve the equation $(8 + 3x)^2 = (5x + 3)^2 - (4x)^2$ , we'll follow these steps:

Step 1: Simplify each side of the equation using known algebraic identities.
Step 2: Solve for $x$ after simplifying the equation.

Step 1: Let's expand each side of the equation.

The left side is $(8 + 3x)^2 = 8^2 + 2 \cdot 8 \cdot 3x + (3x)^2 = 64 + 48x + 9x^2$ .

For the right side, we use the difference of squares identity:

$(5x + 3)^2 - (4x)^2 = \left((5x + 3) - (4x)\right)\left((5x + 3) + (4x)\right)$

Simplifying each:

$(5x + 3) - (4x) = x + 3$

$(5x + 3) + (4x) = 9x + 3$

So, the right side becomes $(x + 3)(9x + 3) = x(9x + 3) + 3(9x + 3) = 9x^2 + 3x + 27x + 9 = 9x^2 + 30x + 9$ .

Now equate the simplified expressions from both sides:

$64 + 48x + 9x^2 = 9x^2 + 30x + 9$

Cancel the $9x^2$ terms from both sides:

$64 + 48x = 30x + 9$

Subtract $30x$ from both sides to isolate the terms:

$64 + 48x - 30x = 9$

$64 + 18x = 9$

Subtract 64 from both sides:

$18x = 9 - 64$

$18x = -55$

Divide both sides by 18 to solve for $x$ :

$x = \frac{-55}{18}$

Simplifying, this gives:

$x = -3 \frac{1}{18}$

The solution to the problem is $x = -3 \frac{1}{18}$ , which matches choice 3.

3

Final Answer

$-3\frac{1}{18}$

Key Points to Remember

Essential concepts to master this topic

Identity: Use $a^2 - b^2 = (a-b)(a+b)$ to simplify equations
Technique: $(5x+3)^2 - (4x)^2 = (x+3)(9x+3)$ factoring first
Check: Substitute $x = -3\frac{1}{18}$ back into original equation ✓

Common Mistakes

Avoid these frequent errors

Expanding all squares before using difference of squares identity
Don't expand $(5x+3)^2 - (4x)^2$ individually = $25x^2 + 30x + 9 - 16x^2$ creates unnecessary complexity! This makes the algebra much harder and increases error chances. Always recognize and apply $a^2 - b^2 = (a-b)(a+b)$ first.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that has the same value as the following:

$$ (x+y)^2 $$

FAQ

Everything you need to know about this question

When should I use the difference of squares formula?

+

Use $a^2 - b^2 = (a-b)(a+b)$ whenever you see two squared terms being subtracted, like $(5x+3)^2 - (4x)^2$ . This simplifies the problem much faster than expanding!

Why didn't we expand the left side using difference of squares?

+

The left side $(8+3x)^2$ is just one square term, not a difference. The difference of squares formula only works when you have subtraction between two squares.

How do I factor (x+3)(9x+3) correctly?

+

Use the FOIL method: First terms: $x \cdot 9x = 9x^2$ , Outer: $x \cdot 3 = 3x$ , Inner: $3 \cdot 9x = 27x$ , Last: $3 \cdot 3 = 9$ . Result: $9x^2 + 30x + 9$

What if I get a mixed number as my answer?

+

Mixed numbers like $-3\frac{1}{18}$ are perfectly valid! Convert to check: $-3\frac{1}{18} = \frac{-55}{18}$ . Always verify by substituting back into the original equation.

Can I solve this by taking square roots of both sides?

+

No! Taking square roots only works when you have $(expression)^2 = number$ . Here we have $(8+3x)^2 = (5x+3)^2 - (4x)^2$ , so the right side isn't just a number.

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Solve (8+3x)² = (5x+3)² - (4x)²: Perfect Square Equation

Perfect Squares with Difference of Squares

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