Expand the Binomial Product: (2x-3)(5x-7) Step-by-Step

Q: Why do I get confused with the negative signs?

Negative signs can be tricky! Remember that negative times positive equals negative, and negative times negative equals positive. In $ (2x-3)(5x-7) $, when you multiply $ (-3) \times (-7) = +21 $.

Q: How do I remember the FOIL order?

F-O-I-L stands for First, Outside, Inside, Last. Draw lines connecting the terms to visualize: First terms together, Outside terms (the outer ones), Inside terms (the inner ones), and Last terms together.

Q: What if I get the wrong middle term?

The middle term comes from adding the Outside and Inside products. In this problem: $ -14x + (-15x) = -29x $. Double-check that you multiplied correctly and combined like terms properly.

Q: Can I use a different method besides FOIL?

Yes! You can use the distributive property by distributing each term in the first binomial to each term in the second. FOIL is just a systematic way to remember this process for binomials.

Q: How do I know my final answer is correct?

Substitute a simple value like $ x = 1 $ into both the original expression and your answer. For $ x = 1 $: $ (2-3)(5-7) = (-1)(-2) = 2 $ and $ 10(1)^2 - 29(1) + 21 = 2 $ ✓

FOIL Method with Negative Terms

Solve the following exercise

$(2x-3)(5x-7)=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solution

00:03 Open parentheses properly, multiply each factor by each factor

00:30 Calculate the products

00:57 Positive times negative always equals negative

01:06 Collect like terms

01:13 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

Solve the following exercise

$(2x-3)(5x-7)=$

Step-by-step solution

To solve the exercise $(2x-3)(5x-7)$ , we must expand the expression by using the distributive property, commonly referred to as the FOIL method for binomials.

First, multiply the first terms of each binomial: $2x \cdot 5x = 10x^2$ .
Outside, multiply the outer terms of the binomials: $2x \cdot -7 = -14x$ .
Inside, multiply the inner terms of the binomials: $-3 \cdot 5x = -15x$ .
Last, multiply the last terms of the binomials: $-3 \cdot -7 = 21$ .

After performing these operations, the expanded expression is:

$10x^2 - 14x - 15x + 21$ .

The next step is to combine the like terms. In this case, the like terms are the linear terms $-14x$ and $-15x$ :

$10x^2 - 14x - 15x + 21 = 10x^2 - 29x + 21$ .

Thus, after simplifying, the expression is $10x^2 - 29x + 21$ .

Therefore, the solution to the expression $(2x-3)(5x-7)$ is $10x^2 - 29x + 21$ .

Final Answer

^{$10x^2-29x+21$}

Key Points to Remember

Essential concepts to master this topic

FOIL: Multiply First, Outside, Inside, Last terms systematically
Technique: $2x \cdot (-7) = -14x$ and $(-3) \cdot 5x = -15x$
Check: Combine like terms: $-14x - 15x = -29x$ ✓

Common Mistakes

Avoid these frequent errors

Adding instead of multiplying with FOIL
Don't add terms like 2x + 5x = 7x when FOILing! This gives completely wrong expressions. The FOIL method requires multiplication at each step: First × First, Outside × Outside, etc. Always multiply, never add during expansion.

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 do I get confused with the negative signs?

Negative signs can be tricky! Remember that negative times positive equals negative, and negative times negative equals positive. In $(2x-3)(5x-7)$ , when you multiply $(-3) \times (-7) = +21$ .

How do I remember the FOIL order?

F-O-I-L stands for First, Outside, Inside, Last. Draw lines connecting the terms to visualize: First terms together, Outside terms (the outer ones), Inside terms (the inner ones), and Last terms together.

What if I get the wrong middle term?

The middle term comes from adding the Outside and Inside products. In this problem: $-14x + (-15x) = -29x$ . Double-check that you multiplied correctly and combined like terms properly.

Can I use a different method besides FOIL?

Yes! You can use the distributive property by distributing each term in the first binomial to each term in the second. FOIL is just a systematic way to remember this process for binomials.

How do I know my final answer is correct?

Substitute a simple value like $x = 1$ into both the original expression and your answer. For $x = 1$ : $(2-3)(5-7) = (-1)(-2) = 2$ and $10(1)^2 - 29(1) + 21 = 2$ ✓

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