Find the Missing Term in the Equation: x² + ? + 9 = (x + 3)²

Q: Why is the middle term 6x and not just 6?

The middle term comes from $ 2ab $ in the formula. Here, $ a = x $ and $ b = 3 $, so $ 2ab = 2 \times x \times 3 = 6x $. The x is essential!

Q: How do I remember the perfect square formula?

Think FOIL backwards! $ (a + b)^2 $ means $ (a + b)(a + b) $. First terms: $ a^2 $, Outer + Inner: $ 2ab $, Last terms: $ b^2 $.

Q: What if I have (x - 3)² instead?

Use $ (a - b)^2 = a^2 - 2ab + b^2 $. The middle term becomes negative: $ (x - 3)^2 = x^2 - 6x + 9 $.

Q: Can I work backwards from the expanded form?

Absolutely! If you see $ x^2 + 6x + 9 $, look for a pattern: first and last terms are perfect squares, and the middle term equals $ 2 \times \sqrt{first} \times \sqrt{last} $.

Q: Why does this equal (x + 3)² exactly?

Because we're finding what makes both sides identical expressions. When we expand $ (x + 3)^2 $, we get $ x^2 + 6x + 9 $. So the missing piece must be $ 6x $ to complete the match!

Perfect Square Trinomials with Missing Terms

Fill in the blanks:

$x^2+\text{?}+9=(x+3)^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing

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

00:10 In this case X is the A

00:14 and 3 is the B

00:18 We'll substitute according to the formula

00:37 We'll identify the difference and complete the unknown

00:41 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

Fill in the blanks:

$x^2+\text{?}+9=(x+3)^2$

Step-by-step solution

To solve this problem, let's start by expanding the expression on the right side of the equation using the formula for the square of a sum.

Step 1: Expand $(x + 3)^2$ :

Using the formula $(a + b)^2 = a^2 + 2ab + b^2$ , we substitute $a = x$ and $b = 3$ .
We have: $(x + 3)^2 = x^2 + 2 \times x \times 3 + 3^2$ .
Simplifying, we get: $x^2 + 6x + 9$ .

Step 2: Compare with the given expression:

The left side of the equation is $x^2 + \text{?} + 9$ .
From our expansion, we see $x^2 + 6x + 9$ matches the structure $x^2 + \text{?} + 9$ .
This implies the missing term $\text{?}$ is $6x$ .

Therefore, the missing term in the expression $x^2 + \text{?} + 9 = (x + 3)^2$ is $6x$ .

Final Answer

$6x$

Key Points to Remember

Essential concepts to master this topic

Formula: $(a + b)^2 = a^2 + 2ab + b^2$ expansion pattern
Technique: Expand $(x + 3)^2 = x^2 + 6x + 9$ using the formula
Check: Verify $x^2 + 6x + 9 = (x + 3)^2$ by expanding ✓

Common Mistakes

Avoid these frequent errors

Forgetting the middle term coefficient
Don't think $(x + 3)^2 = x^2 + 9$ and miss the 6x term! This ignores the $2ab$ part of the formula and gives an incomplete expansion. Always include all three terms: $a^2 + 2ab + b^2$ .

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 is the middle term 6x and not just 6?

The middle term comes from $2ab$ in the formula. Here, $a = x$ and $b = 3$ , so $2ab = 2 \times x \times 3 = 6x$ . The x is essential!

How do I remember the perfect square formula?

Think FOIL backwards! $(a + b)^2$ means $(a + b)(a + b)$ . First terms: $a^2$ , Outer + Inner: $2ab$ , Last terms: $b^2$ .

What if I have (x - 3)² instead?

Use $(a - b)^2 = a^2 - 2ab + b^2$ . The middle term becomes negative: $(x - 3)^2 = x^2 - 6x + 9$ .

Can I work backwards from the expanded form?

Absolutely! If you see $x^2 + 6x + 9$ , look for a pattern: first and last terms are perfect squares, and the middle term equals $2 \times \sqrt{first} \times \sqrt{last}$ .

Why does this equal (x + 3)² exactly?

Because we're finding what makes both sides identical expressions. When we expand $(x + 3)^2$ , we get $x^2 + 6x + 9$ . So the missing piece must be $6x$ to complete the match!

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