Solve (x+1)² + (x+2)²: Sum of Squared Binomial Expressions

Q: Why can't I just square the sum like (x+1+x+2)²?

Because squaring a sum is different from summing squares! The expression (x+1)² + (x+2)² means two separate squared terms being added, not one large squared expression.

Q: How do I remember the binomial expansion formula?

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

Q: What if I get different numbers in the binomials?

The same method works! Just apply $ (a+b)^2 = a^2 + 2ab + b^2 $ to each binomial separately, then add all terms together. Always combine like terms at the end.

Expanding Binomials with Sum Operations

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:07 We'll use the shortened multiplication formulas to open the parentheses

00:35 We'll calculate the multiplications and squares

00:41 We'll collect like terms

00:57 We'll factor out the common term from the parentheses

00:59 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

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

Step-by-step solution

In order to solve the exercise, we will need to know the abbreviated multiplication formula:

In this exercise, we will use the formula twice:

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

$(x+2)^2=x^2+4x+4$

Now, we add:

$x^2+2x+1+x^2+4x+4=2x^2+6x+5$

x²+2x+1+x²+4x+4=
2x²+6x+5

Note that a common factor can be extracted from part of the digits: $2(x^2+3x)+5$

Final Answer

$2(x^2+3x)+5$

Key Points to Remember

Essential concepts to master this topic

Formula: Use $(a+b)^2 = a^2 + 2ab + b^2$ for each binomial
Technique: Expand separately: $(x+1)^2 = x^2+2x+1$ and $(x+2)^2 = x^2+4x+4$
Check: Combine like terms: $2x^2 + 6x + 5$ factors to $2(x^2+3x) + 5$ ✓

Common Mistakes

Avoid these frequent errors

Adding the expressions before expanding
Don't try to combine (x+1)² + (x+2)² = (x+3)² = x² + 6x + 9! This shortcut doesn't exist and gives the wrong answer. Always expand each squared binomial separately using the formula, then add the results.

Practice Quiz

Test your knowledge with interactive questions

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

$$ (x+3)^2 $$

FAQ

Everything you need to know about this question

Why can't I just square the sum like (x+1+x+2)²?

Because squaring a sum is different from summing squares! The expression (x+1)² + (x+2)² means two separate squared terms being added, not one large squared expression.

Do I always get the same middle term coefficient?

No! The middle term depends on the numbers in your binomials. For (x+1)² you get 2x, for (x+2)² you get 4x. Always calculate each expansion carefully.

How do I remember the binomial expansion formula?

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

Should I factor my final answer?

It depends on what the question asks! Here we can factor out 2 from some terms: $2x^2 + 6x + 5 = 2(x^2 + 3x) + 5$ , but the standard form $2x^2 + 6x + 5$ is usually fine.

What if I get different numbers in the binomials?

The same method works! Just apply $(a+b)^2 = a^2 + 2ab + b^2$ to each binomial separately, then add all terms together. Always combine like terms at the end.

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