Solve (3x+4)(5+?) = 15x+3xy+4y+20: Find the Missing Term

Q: How do I know if the missing term is a variable or number?

Look at the expanded form on the right side! If you see terms with variables (like 3xy and 4y), the missing term likely contains those same variables.

Q: What if I expand and get different terms than shown?

This means your missing term is wrong. Go back and compare each term carefully - the coefficients and variables must match exactly on both sides.

Q: Why do both 3x? and 4? give me the same answer?

This is your verification! When you get the same value for ? from different terms, it confirms your answer is correct. If they gave different values, you'd know something was wrong.

Q: What if I can't figure out what ? should be?

Start by expanding the left side step by step using $ (a+b)(c+d) = ac + ad + bc + bd $. Then line up each term with the right side - the pattern will become clear!

Algebraic Factorization with Missing Variables

Fill in the missing number

$(3x+4)(5+?)=15x+3xy+4y+20$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing number

00:03 Let's use N as the unknown

00:06 Let's properly open parentheses, multiply each factor by each factor

00:27 Let's calculate the products

00:48 Let's compare the corresponding expressions

00:55 Let's reduce what we can

01:00 This is the value of N, now let's check if it fits

01:05 Let's compare the corresponding expressions, substitute the N value we found

01:10 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 missing number

$(3x+4)(5+?)=15x+3xy+4y+20$

Step-by-step solution

To solve this problem, we'll proceed as follows:

Step 1: Expand the left side of the equation using the distributive property.
Step 2: Compare the expanded expression with the right side.
Step 3: Solve for the missing number to make the expressions equal.

Now, let's work through these steps:
Step 1: Apply the distributive property to expand $(3x + 4)(5 + ?)$ :
$(3x + 4)(5 + ?) = 3x(5 + ?) + 4(5 + ?)$ .
Expanding each term, we get:
$= 3x \cdot 5 + 3x \cdot ? + 4 \cdot 5 + 4 \cdot ?$ ,
which simplifies to $15x + 3x? + 20 + 4?$ .

Step 2: Compare this with the right side, $15x + 3xy + 4y + 20$ .

Step 3: By comparing terms, note:
- $15x$ matches on both sides.
- $20$ matches on both sides.
- $3x? = 3xy$ implies that $? = y$ .
- $4? = 4y$ confirms $? = y$ .

Therefore, the correct missing number that should replace the question mark is $\boxed{y}$ .

Final Answer

$y$

Key Points to Remember

Essential concepts to master this topic

Distributive Property: Expand (a+b)(c+d) = ac + ad + bc + bd systematically
Term Matching: Compare coefficients: 3x? = 3xy means ? = y
Double Check: Verify both 3x? = 3xy and 4? = 4y give same result ✓

Common Mistakes

Avoid these frequent errors

Assuming the missing term must be a number
Don't assume ? must be a constant like 2 or 5 = wrong expansion! This ignores that algebraic expressions can contain variables. Always consider that the missing term could be a variable like y or an expression like xy.

Practice Quiz

Test your knowledge with interactive questions

$(3+20)\times(12+4)=$

FAQ

Everything you need to know about this question

How do I know if the missing term is a variable or number?

Look at the expanded form on the right side! If you see terms with variables (like 3xy and 4y), the missing term likely contains those same variables.

What if I expand and get different terms than shown?

This means your missing term is wrong. Go back and compare each term carefully - the coefficients and variables must match exactly on both sides.

Can the missing term be something like 2y or xy?

Absolutely! The missing term can be any algebraic expression. Use the distributive property to expand, then match terms to find what works.

Why do both 3x? and 4? give me the same answer?

This is your verification! When you get the same value for ? from different terms, it confirms your answer is correct. If they gave different values, you'd know something was wrong.

What if I can't figure out what ? should be?

Start by expanding the left side step by step using $(a+b)(c+d) = ac + ad + bc + bd$ . Then line up each term with the right side - the pattern will become clear!

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