Solve (2a+b)(3b-?) = 6ab+5a+2b: Find the Missing Term

Polynomial Distribution with Missing Terms

Fills in the missing element

(2a+b)(3b?)=6ab+5a+2b (2a+b)(3b-?)=6ab+5a+2b

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Complete the missing term
00:04 Let X be the unknown
00:11 Open parentheses properly, multiply each factor by each factor
00:40 Move terms to the other side
00:56 Collect like terms
01:01 Factor out the common term
01:15 Isolate the unknown X
01:22 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Fills in the missing element

(2a+b)(3b?)=6ab+5a+2b (2a+b)(3b-?)=6ab+5a+2b

2

Step-by-step solution

Let's solve this problem step by step.

Step 1: Apply the distributive property to the left-hand side.
The expression (2a+b)(3b?)(2a+b)(3b-?) expands to:

(2a+b)(3b?)=2a(3b?)+b(3b?) (2a + b)(3b - ?) = 2a(3b - ?) + b(3b - ?)

Step 2: Distribute each term:

  • First 2a(3b)2a(3b) gives 6ab6ab.
  • Then 2a(?)2a(-?) gives 2a?-2a\cdot?.
  • Next b(3b)b(3b) gives 3b23b^2.
  • Finally b(?)b(-?) gives b?-b\cdot?.

In equation form:

=6ab+3b22a?b? = 6ab + 3b^2 - 2a? - b?

Now, compare this with the given equation: 6ab+5a+2b6ab + 5a + 2b.

Notice that 3b23b^2 doesn't present on the right-hand side, indicating no term in b2b^2 should exist. Moreover, right-hand terms 5a5a and 2b2b have no zero counterparts on complex variables (polynomial formation), indicating inconsistencies in all forms with possible simple polynomial inequivalence.

Step 3: Analysis shows inability to correlate purely intuitively with missing match coefficients specifically implies an inadequacy here in probabilistic assumptions, leading us to:

Therefore, the solution to the problem with the provided options is No adequate solution.

3

Final Answer

No adequate solution

Key Points to Remember

Essential concepts to master this topic
  • Distribution Rule: Each term multiplies every term in the other factor
  • Technique: Compare coefficients systematically: 6ab+3b22a?b? 6ab + 3b^2 - 2a? - b? vs 6ab+5a+2b 6ab + 5a + 2b
  • Check: All terms on both sides must match exactly for a valid solution ✓

Common Mistakes

Avoid these frequent errors
  • Ignoring coefficient mismatch between expanded and given forms
    Don't assume a solution exists just because the problem asks for one = impossible equation! The expanded form produces 3b2 3b^2 which doesn't appear on the right side, making the equation inconsistent. Always compare all terms systematically to determine if a solution exists.

Practice Quiz

Test your knowledge with interactive questions

It is possible to use the distributive property to simplify the expression below?

What is its simplified form?

\( (ab)(c d) \)

\( \)

FAQ

Everything you need to know about this question

Why doesn't this equation have a solution?

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When we expand (2a+b)(3b?) (2a+b)(3b-?) , we get a 3b2 3b^2 term that cannot be eliminated no matter what we substitute for the missing value. Since the right side has no b2 b^2 term, the equation is impossible.

How do I know when to look for 'No solution'?

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After expanding and simplifying, compare the structure of both sides. If one side has terms (like b2 b^2 ) that the other side cannot possibly have, then no solution exists.

What if I tried to solve for the missing term anyway?

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You'd get contradictory equations! For example, trying to match coefficients would require 3b2=0 3b^2 = 0 for all values of b, which is impossible.

Could there be a typo in the original problem?

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Possibly! If the right side were 6ab+3b2+5a+2b 6ab + 3b^2 + 5a + 2b instead, then we could find the missing term. Always work with what's given and recognize when no solution exists.

How is this different from regular distribution problems?

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Most distribution problems have solutions, but this one tests your ability to recognize impossible equations. It's important to know that not all mathematical problems have solutions!

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