Verify the Equation: (-2a+3b)(4c+5a) = 8ac+10a²-12bc-15ab

Q: Why are all the signs opposite in my answer?

This means you calculated correctly! When expanding $ (-2a+3b)(4c+5a) $, you get $ -8ac-10a^2+12bc+15ab $, which is exactly the negative of the given expression.

Q: How do I remember which terms to multiply together?

Use FOIL: First terms, Outer terms, Inner terms, Last terms. For $ (-2a+3b)(4c+5a) $:First: $ (-2a)(4c) = -8ac $Outer: $ (-2a)(5a) = -10a^2 $Inner: $ (3b)(4c) = 12bc $Last: $ (3b)(5a) = 15ab $

Q: What does it mean when the expression equals the negative?

It means the given equation is incorrect. The left side equals $ -(\text{right side}) $. This is a common type of verification problem where you check if an algebraic identity is true or false.

Q: Should I always expand the left side completely?

Yes! Always expand fully and combine like terms before comparing. Don't try to match terms before completing all the multiplication - you might miss sign errors or incorrectly combine terms.

Q: How can I avoid sign errors when multiplying?

Write out each multiplication step clearly:Positive × Positive = PositiveNegative × Positive = NegativePositive × Negative = NegativeNegative × Negative = PositiveTake your time with each step!

Polynomial Multiplication with Sign Verification

Is equality correct?

$(-2a+3b)(4c+5a)\stackrel{?}{=}8ac+10a^2-12bc-15ab$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Are the expressions equal?

00:04 Let's properly open parentheses, multiply each factor by each factor

00:24 Let's calculate the multiplications

00:47 We'll take out the minus and see that then the expressions are equal, therefore they are not equal

00:54 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

Is equality correct?

$(-2a+3b)(4c+5a)\stackrel{?}{=}8ac+10a^2-12bc-15ab$

Step-by-step solution

To determine if the given algebraic expression is correct, we will expand the left-hand side using the distributive property:

The expression is $(-2a + 3b)(4c + 5a)$ .

Step-by-step expansion:

Multiply $-2a$ by $4c$ : $-2a \times 4c = -8ac$ .
Multiply $-2a$ by $5a$ : $-2a \times 5a = -10a^2$ .
Multiply $3b$ by $4c$ : $3b \times 4c = 12bc$ .
Multiply $3b$ by $5a$ : $3b \times 5a = 15ab$ .

Combine these results: $-8ac - 10a^2 + 12bc + 15ab$ .

Now compare this result with the right-hand side of the given expression $8ac + 10a^2 - 12bc - 15ab$ .

We can observe that each corresponding term has the opposite sign.

This shows that the original statement is incorrect.

Therefore, the expression is actually the negative of what was given, so:

The expression is exactly the same as $-(8ac + 10a^2 - 12bc - 15ab)$ .

Thus, the correct choice is:

No, the expression is exactly the same as $-(8ac + 10a^2 - 12bc - 15ab)$ .

Final Answer

No, the expression is exactly the same as $-(8ac+10a^2-12bc-15ab)$

Key Points to Remember

Essential concepts to master this topic

FOIL Method: Distribute each term to get all four products
Technique: $(-2a)(4c) = -8ac$ and $(3b)(5a) = 15ab$
Check: Compare each term's sign in your result with given expression ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute negative signs correctly
Don't forget that $(-2a)(4c) = -8ac$ , not $8ac$ ! Missing negative signs gives completely opposite results. Always track each negative sign through every multiplication step.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why are all the signs opposite in my answer?

This means you calculated correctly! When expanding $(-2a+3b)(4c+5a)$ , you get $-8ac-10a^2+12bc+15ab$ , which is exactly the negative of the given expression.

How do I remember which terms to multiply together?

Use FOIL: First terms, Outer terms, Inner terms, Last terms. For $(-2a+3b)(4c+5a)$ :

First: $(-2a)(4c) = -8ac$
Outer: $(-2a)(5a) = -10a^2$
Inner: $(3b)(4c) = 12bc$
Last: $(3b)(5a) = 15ab$

What does it mean when the expression equals the negative?

It means the given equation is incorrect. The left side equals $-(\text{right side})$ . This is a common type of verification problem where you check if an algebraic identity is true or false.

Should I always expand the left side completely?

Yes! Always expand fully and combine like terms before comparing. Don't try to match terms before completing all the multiplication - you might miss sign errors or incorrectly combine terms.

How can I avoid sign errors when multiplying?

Write out each multiplication step clearly:

Positive × Positive = Positive
Negative × Positive = Negative
Positive × Negative = Negative
Negative × Negative = Positive

Take your time with each step!

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