Perfect Square Expansion: Find the Missing Term in x² + ? + 9 = (x-3)²

Q: Why is the middle term negative when we have (x-3)²?

Because of the binomial formula! When you expand $ (x-3)^2 $, you get $ x^2 - 2(x)(3) + 3^2 $. The middle term $ -2(x)(3) = -6x $ is negative.

Q: How do I remember the binomial expansion formula?

Use the pattern: First² - 2(First)(Last) + Last². For $ (x-3)^2 $: x² minus 2(x)(3) plus 3² = $ x^2 - 6x + 9 $.

Q: What if it was (x+3)² instead?

Then the middle term would be positive! $ (x+3)^2 = x^2 + 6x + 9 $ because $ +2(x)(3) = +6x $. The sign in the binomial determines the middle term's sign.

Q: Can I solve this by working backwards from the answer choices?

Yes! Substitute each choice into $ x^2 + ? + 9 $ and see which one equals $ (x-3)^2 $ when expanded. But understanding the binomial formula is more reliable!

Q: Why do both sides have the same constant term (9)?

Because $ 3^2 = 9 $! In any perfect square trinomial $ (x+k)^2 $, the constant term is always $ k^2 $. Since we have 9, we know $ k = ±3 $.

Binomial Expansion with Missing Middle Terms

Fill in the blanks:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's fill in the missing parts!

00:10 We will use shortened multiplication formulas, to expand the parentheses.

00:20 Next, we will solve the multiplication step by step.

00:25 Let's find what's missing, and complete it!

00:28 And that's how we solve this problem. Great job!

Step-by-step written solution

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Understand the problem

Fill in the blanks:

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

Step-by-step solution

To address this mathematical problem, we will apply the square of a binomial formula and solve for the missing term. Here's how:

Step 1: Expand the right-hand side of the equation $(x-3)^2$ .
Step 2: Equate the expanded form to $x^2 + ? + 9$ .
Step 3: Solve for the missing term by comparing the coefficients.

Step 1: Expanding $(x-3)^2$ using the formula, we get:

$(x - 3)^2 = x^2 - 2 \cdot x \cdot 3 + 3^2$ .

This simplifies to:

$x^2 - 6x + 9$ .

Step 2: Equating this to the left-hand side:

$x^2 + ? + 9 = x^2 - 6x + 9$ .

Step 3: Compare the terms:

The term that replaces "?" on the left-hand side must make the equation hold.

Setting corresponding terms equal, we find that:

$? = -6x$ .

Therefore, the solution to the problem is $-6x$ .

Final Answer

$-6x$

Key Points to Remember

Essential concepts to master this topic

Formula: $(a-b)^2 = a^2 - 2ab + b^2$ pattern
Technique: Expand $(x-3)^2 = x^2 - 6x + 9$ then match coefficients
Check: Verify $x^2 - 6x + 9 = (x-3)^2$ by expanding back ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative sign in the middle term
Don't expand $(x-3)^2$ as $x^2 + 6x + 9$ = wrong middle term! The negative in (x-3) makes the middle term negative: -2(x)(3) = -6x. Always use the correct binomial formula $(a-b)^2 = a^2 - 2ab + b^2$ .

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 is the middle term negative when we have (x-3)²?

Because of the binomial formula! When you expand $(x-3)^2$ , you get $x^2 - 2(x)(3) + 3^2$ . The middle term $-2(x)(3) = -6x$ is negative.

How do I remember the binomial expansion formula?

Use the pattern: First² - 2(First)(Last) + Last². For $(x-3)^2$ : x² minus 2(x)(3) plus 3² = $x^2 - 6x + 9$ .

What if it was (x+3)² instead?

Then the middle term would be positive! $(x+3)^2 = x^2 + 6x + 9$ because $+2(x)(3) = +6x$ . The sign in the binomial determines the middle term's sign.

Can I solve this by working backwards from the answer choices?

Yes! Substitute each choice into $x^2 + ? + 9$ and see which one equals $(x-3)^2$ when expanded. But understanding the binomial formula is more reliable!

Why do both sides have the same constant term (9)?

Because $3^2 = 9$ ! In any perfect square trinomial $(x+k)^2$ , the constant term is always $k^2$ . Since we have 9, we know $k = ±3$ .

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