Verify Equivalence: 5x²+7x+7 = (2x+3)(3x+4)

Q: Do I have to expand both sides to check equivalence?

No! Only expand the factored side $ (2x+3)(3x+4) $. The left side $ 5x^2+7x+7 $ is already in standard form, so just compare coefficients directly.

Q: What's the fastest way to multiply two binomials?

Use FOIL: First terms, Outer terms, Inner terms, Last terms. For $ (2x+3)(3x+4) $: F: 2x·3x, O: 2x·4, I: 3·3x, L: 3·4. Then combine like terms!

Q: Why do all the coefficients have to match exactly?

Because polynomials are identical only when every corresponding term matches perfectly. If even one coefficient differs, like $ 6x^2 ≠ 5x^2 $, the expressions represent different values.

Q: Can I check by plugging in a number for x?

That's not reliable for proving equivalence! Two different polynomials might give the same result for one specific x-value but differ for others. Always compare coefficients instead.

Polynomial Expansion with Coefficient Comparison

Are the expressions on both sides equivalent?

$5x^2+7x+7\stackrel{?}{=}(2x+3)(3x+4)$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Are these expressions equal? Let's find out.

00:15 First, we open the parentheses. Multiply each term carefully. Take your time!

00:28 Great job! Now, calculate the products for each multiplication.

00:52 Next, let's group the factors together to simplify.

00:56 Now, compare each term closely to see the differences.

01:01 Since all terms differ, the expressions are not equal.

01:05 And there you have it! That's how we solve this problem.

Step-by-step written solution

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Understand the problem

Are the expressions on both sides equivalent?

$5x^2+7x+7\stackrel{?}{=}(2x+3)(3x+4)$

Step-by-step solution

To determine if the expressions are equivalent, we need to expand the right-side expression, $(2x + 3)(3x + 4)$ , and compare it with $5x^2 + 7x + 7$ .

Let's expand the right-side expression:

First, use the distributive property (or FOIL):
$(2x + 3)(3x + 4) = 2x \cdot 3x + 2x \cdot 4 + 3 \cdot 3x + 3 \cdot 4$
This simplifies to $6x^2 + 8x + 9x + 12$ .
Combine like terms: $6x^2 + (8x + 9x) + 12 = 6x^2 + 17x + 12$ .

Now, compare the expanded expression $6x^2 + 17x + 12$ to the left side $5x^2 + 7x + 7$ :

The coefficient of $x^2$ is 6 on the right, but 5 on the left.
The coefficient of $x$ is 17 on the right, but 7 on the left.
The constant term is 12 on the right, but 7 on the left.

Since all corresponding coefficients differ between the two sides, the expressions are not equivalent.

Therefore, the correct answer is: No, because all the coefficients of the corresponding terms in the expressions on both sides of the equation are different.

Final Answer

No, because all the coefficients of the corresponding terms in the expressions on both sides of the equation are different.

Key Points to Remember

Essential concepts to master this topic

Expansion Rule: Use FOIL to multiply (2x+3)(3x+4) completely
Technique: First: 2x·3x = 6x², Outer: 2x·4 = 8x, Inner: 3·3x = 9x, Last: 3·4 = 12
Check: Compare all coefficients: x² terms (6≠5), x terms (17≠7), constants (12≠7) ✓

Common Mistakes

Avoid these frequent errors

Assuming expressions are equal without expanding
Don't just look at the factored form and guess they're equal = wrong conclusion! The factored form hides the actual coefficients. Always expand the right side completely and compare each coefficient term by term.

Practice Quiz

Test your knowledge with interactive questions

$$ (x+y)(x-y)= $$

FAQ

Everything you need to know about this question

Do I have to expand both sides to check equivalence?

No! Only expand the factored side $(2x+3)(3x+4)$ . The left side $5x^2+7x+7$ is already in standard form, so just compare coefficients directly.

What's the fastest way to multiply two binomials?

Use FOIL: First terms, Outer terms, Inner terms, Last terms. For $(2x+3)(3x+4)$ : F: 2x·3x, O: 2x·4, I: 3·3x, L: 3·4. Then combine like terms!

Why do all the coefficients have to match exactly?

Because polynomials are identical only when every corresponding term matches perfectly. If even one coefficient differs, like $6x^2 ≠ 5x^2$ , the expressions represent different values.

Can I check by plugging in a number for x?

That's not reliable for proving equivalence! Two different polynomials might give the same result for one specific x-value but differ for others. Always compare coefficients instead.

What if I make an arithmetic error during expansion?

Double-check each step! Verify: 2x·3x = 6x², 8x + 9x = 17x, and that you didn't miss any terms. Careful arithmetic is crucial for accurate coefficient comparison.

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