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

\( (x+y)(x-y)= \)

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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