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

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

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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