Solve (x-1)² - (x+2)² = 15: Difference of Squared Binomials

Q: Why don't I just expand both squares separately first?

You absolutely should! Expanding $ (x-1)^2 $ and $ (x+2)^2 $ separately using the perfect square formula is the correct first step. Then subtract the second from the first.

Q: What happens to the x² terms in this problem?

The $ x^2 $ terms cancel out completely! You get $ x^2 - x^2 = 0 $, which means this becomes a linear equation instead of quadratic.

Q: How do I remember the perfect square formula?

Think "First, Outer, Inner, Last" or remember the pattern: $ (a+b)^2 = a^2 + 2ab + b^2 $$ (a-b)^2 = a^2 - 2ab + b^2 $ The middle term is always twice the product of the two terms.

Q: Why is the answer negative when both squares are positive?

Good observation! While both $ (x-1)^2 $ and $ (x+2)^2 $ are positive, we're looking at their difference. When $ x = -3 $, the second square is larger, making the overall result positive.

Q: Can I use the difference of squares formula instead?

Not directly! The difference of squares formula $ a^2 - b^2 = (a+b)(a-b) $ applies when you have perfect squares being subtracted. Here you have binomial squares, so expand them first.

Difference of Squares with Distributive Property

$(x-1)^2-(x+2)^2=15$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 We'll use the shortened multiplication formulas to open all parentheses

00:21 Negative times positive is always negative

00:27 Negative times negative is always positive

00:35 Group like terms

00:49 Isolate X

01:08 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(x-1)^2-(x+2)^2=15$

Step-by-step solution

Let's solve the equation, first we'll simplify the algebraic expressions using the perfect square binomial formula:

$(a\pm b)^2=a^2\pm2ab+b^2$ We'll apply the mentioned formula and expand the parentheses in the expressions in the equation:

$(x-1)^2-(x+2)^2=15 \\ x^2-2\cdot x\cdot1+1^2-(x^2+2\cdot x\cdot2+2^2)=15 \\ x^2-2x+1-(x^2+4x+4)=15\\ x^2-2x+1-x^2-4x-4=15$ In the final stage, we used the distributive property to expand the parentheses,

We'll continue and combine like terms, by moving terms between sides, later - we can notice that the squared term cancels out and therefore it's a first-degree equation, which we'll solve by isolating the variable term on one side and dividing both sides of the equation by its coefficient:

$x^2-2x+1-x^2-4x-4=15 \\ -6x=18\hspace{8pt}\text{/}:(-6)\\ \boxed{x=-3}$ Therefore, the correct answer is answer B.

Final Answer

$x=-3$

Key Points to Remember

Essential concepts to master this topic

Formula: Use $(a-b)^2 = a^2 - 2ab + b^2$ to expand binomials
Technique: Apply distributive property: $-(x^2 + 4x + 4) = -x^2 - 4x - 4$
Check: Substitute $x = -3$ : $(-4)^2 - (1)^2 = 16 - 1 = 15$ ✓

Common Mistakes

Avoid these frequent errors

Incorrectly distributing the negative sign
Don't forget to distribute the negative sign to ALL terms when expanding $-(x^2 + 4x + 4)$ = wrong signs on coefficients! This leads to incorrect combining of like terms and wrong final answers. Always apply the negative sign to each term: $-x^2 - 4x - 4$ .

Practice Quiz

Test your knowledge with interactive questions

$$ (4b-3)(4b-3) $$

Rewrite the above expression as an exponential summation expression:

FAQ

Everything you need to know about this question

Why don't I just expand both squares separately first?

You absolutely should! Expanding $(x-1)^2$ and $(x+2)^2$ separately using the perfect square formula is the correct first step. Then subtract the second from the first.

What happens to the x² terms in this problem?

The $x^2$ terms cancel out completely! You get $x^2 - x^2 = 0$ , which means this becomes a linear equation instead of quadratic.

How do I remember the perfect square formula?

Think "First, Outer, Inner, Last" or remember the pattern:

$(a+b)^2 = a^2 + 2ab + b^2$
$(a-b)^2 = a^2 - 2ab + b^2$

The middle term is always twice the product of the two terms.

Why is the answer negative when both squares are positive?

Good observation! While both $(x-1)^2$ and $(x+2)^2$ are positive, we're looking at their difference. When $x = -3$ , the second square is larger, making the overall result positive.

Can I use the difference of squares formula instead?

Not directly! The difference of squares formula $a^2 - b^2 = (a+b)(a-b)$ applies when you have perfect squares being subtracted. Here you have binomial squares, so expand them first.

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Solve (x-1)² - (x+2)² = 15: Difference of Squared Binomials

Difference of Squares with Distributive Property

❤️ 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 don't I just expand both squares separately first?

What happens to the x² terms in this problem?

How do I remember the perfect square formula?

Why is the answer negative when both squares are positive?

Can I use the difference of squares formula instead?

🌟 Unlock Your Math Potential

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