Solve (x-2)² + (x-3)² = : Expanding Square Binomials

Q: Why can't I just square each part separately like (x-2)² = x² - 2²?

That's not how squaring binomials works! When you square (x-2), you're multiplying (x-2) × (x-2), which gives three terms: x², the middle term -4x, and 4. You can't ignore the middle term!

Q: How do I remember the binomial square formula?

Think of it as "First² - 2×First×Last + Last²". For (x-3)²: x² - 2(x)(3) + 3² = x² - 6x + 9. The middle term is always twice the product of the two terms!

Q: What's the fastest way to combine like terms at the end?

Group by powers of x: collect all x² terms (1x² + 1x² = 2x²), all x terms (-4x - 6x = -10x), and all constants (4 + 9 = 13).

Q: Can I use FOIL method instead of the formula?

Absolutely! FOIL gives the same result: (x-2)(x-2) = x² - 2x - 2x + 4 = x² - 4x + 4. Both methods work, but the formula (x-y)² = x² - 2xy + y² is often faster.

Binomial Expansion with Sum of Squares

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

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

00:11 X is the A in the formula

00:17 And 2 is the B in the formula

00:30 We'll use the same formula where X is A and 3 is B

00:40 Let's solve the multiplications

00:53 Let's group the factors

01:06 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

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

Step-by-step solution

In order to solve the question, we need to know one of the shortcut multiplication formulas:

$(x−y)^2=x^2−2xy+y^2$

We apply the formula twice:

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

$(x-3)^2=x^2-6x+9$

Now we add the two together:

$x^2-4x+4+x^2-6x+9=$

$2 x^2-10x+13$

Final Answer

$2x^2-10x+13$

Key Points to Remember

Essential concepts to master this topic

Formula: Use (x-y)² = x² - 2xy + y² for each binomial
Technique: Expand (x-2)² = x² - 4x + 4, then (x-3)² = x² - 6x + 9
Check: Combine like terms: 2x² + (-4x-6x) + (4+9) = 2x² - 10x + 13 ✓

Common Mistakes

Avoid these frequent errors

Using (x-y)² = x² - y² instead of the complete formula
Don't use (x-2)² = x² - 4 and (x-3)² = x² - 9 = 2x² - 13! This misses the middle term completely and gives a wrong answer. Always use the full formula (x-y)² = x² - 2xy + y² to include all three terms.

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 part separately like (x-2)² = x² - 2²?

That's not how squaring binomials works! When you square (x-2), you're multiplying (x-2) × (x-2), which gives three terms: x², the middle term -4x, and 4. You can't ignore the middle term!

How do I remember the binomial square formula?

Think of it as "First² - 2×First×Last + Last²". For (x-3)²: x² - 2(x)(3) + 3² = x² - 6x + 9. The middle term is always twice the product of the two terms!

What's the fastest way to combine like terms at the end?

Group by powers of x: collect all x² terms (1x² + 1x² = 2x²), all x terms (-4x - 6x = -10x), and all constants (4 + 9 = 13).

Can I use FOIL method instead of the formula?

Absolutely! FOIL gives the same result: (x-2)(x-2) = x² - 2x - 2x + 4 = x² - 4x + 4. Both methods work, but the formula (x-y)² = x² - 2xy + y² is often faster.

How do I check my final answer is correct?

Pick a simple value like x = 0 and substitute into both the original expression (0-2)² + (0-3)² = 4 + 9 = 13 and your answer 2(0)² - 10(0) + 13 = 13. They should match!

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