Simplify: 2(3x-1)² - 3(2x+1)² Expression with Squared Binomials

Q: Why can't I just square each term separately?

Squaring a binomial like $ (3x-1)^2 $ is not the same as squaring each term! You must include the middle term from the formula $ (a-b)^2 = a^2 - 2ab + b^2 $.

Q: How do I remember the binomial square formulas?

Think of FOIL but with identical binomials: $ (a+b)^2 = (a+b)(a+b) $. First + Outer + Inner + Last gives you $ a^2 + ab + ab + b^2 = a^2 + 2ab + b^2 $!

Q: What's the difference between (3x-1)² and 3x-1²?

Huge difference! $ (3x-1)^2 $ means the entire binomial is squared, while $ 3x-1^2 $ means only the 1 is squared. Always use parentheses to show what's being squared!

Q: Why do we need to factor the final answer?

Factoring $ 6x^2 - 24x - 1 $ into $ 6x(x-4) - 1 $ makes it match the answer choices! It's also the simplest form that shows the mathematical structure clearly.

Q: How can I check if my expansion is correct?

Expand your final answer back to standard formCompare coefficients of like termsSubstitute a simple value like x=1 into both original and final expressions

Algebraic Expansion with Binomial Squares

$2(3x-1)^2-3(2x+1)^2=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:03 Use shortened multiplication formulas to open the parentheses

00:20 Square each factor in the multiplication

00:28 Use shortened multiplication formulas to open the parentheses

00:40 Open parentheses properly, multiply by each factor

00:51 Square each factor in the multiplication

00:54 Calculate the multiplication

01:06 Open parentheses properly, multiply by each factor

01:18 Group the factors

01:33 Take out the common factor from the parentheses

01:36 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

$2(3x-1)^2-3(2x+1)^2=$

Step-by-step solution

To solve the expression $2(3x-1)^2 - 3(2x+1)^2$ , we perform these steps:

First, expand and simplify $(3x-1)^2$ :
$(3x-1)^2 = (3x)^2 - 2(3x)(1) + 1^2 = 9x^2 - 6x + 1$ .
Next, expand and simplify $(2x+1)^2$ :
$(2x+1)^2 = (2x)^2 + 2(2x)(1) + 1^2 = 4x^2 + 4x + 1$ .
Now, multiply these by their corresponding coefficients and subtract:
$2(3x-1)^2 = 2(9x^2 - 6x + 1) = 18x^2 - 12x + 2$ ,
$3(2x+1)^2 = 3(4x^2 + 4x + 1) = 12x^2 + 12x + 3$ .
Subtract the second expression from the first:
$18x^2 - 12x + 2 - (12x^2 + 12x + 3) = 18x^2 - 12x + 2 - 12x^2 - 12x - 3$ .
Simplify the expression:
$6x^2 - 24x - 1$ is obtained by combining like terms.
The final expression simplifies to $6x(x-4) - 1$ by factoring out $6x$ .

The correct answer is $\mathbf{6x(x-4)-1}$ , which corresponds to choice 3.

Final Answer

$6x(x-4)-1$

Key Points to Remember

Essential concepts to master this topic

Rule: Use (a±b)² = a² ± 2ab + b² formula
Technique: Factor out common terms: 6x² - 24x - 1 = 6x(x-4) - 1
Check: Expand final answer back to verify: 6x(x-4) - 1 = 6x² - 24x - 1 ✓

Common Mistakes

Avoid these frequent errors

Incorrectly expanding squared binomials
Don't forget the middle term when expanding (3x-1)² = 9x² + 1! This ignores the -2(3x)(1) = -6x term and gives wrong coefficients. Always use the complete formula (a-b)² = a² - 2ab + b².

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 can't I just square each term separately?

Squaring a binomial like $(3x-1)^2$ is not the same as squaring each term! You must include the middle term from the formula $(a-b)^2 = a^2 - 2ab + b^2$ .

How do I remember the binomial square formulas?

Think of FOIL but with identical binomials: $(a+b)^2 = (a+b)(a+b)$ . First + Outer + Inner + Last gives you $a^2 + ab + ab + b^2 = a^2 + 2ab + b^2$ !

What's the difference between (3x-1)² and 3x-1²?

Huge difference! $(3x-1)^2$ means the entire binomial is squared, while $3x-1^2$ means only the 1 is squared. Always use parentheses to show what's being squared!

Why do we need to factor the final answer?

Factoring $6x^2 - 24x - 1$ into $6x(x-4) - 1$ makes it match the answer choices! It's also the simplest form that shows the mathematical structure clearly.

How can I check if my expansion is correct?

Expand your final answer back to standard form
Compare coefficients of like terms
Substitute a simple value like x=1 into both original and final expressions

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Short Multiplication Formulas questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime