(7x-2y)^2 Алгебра 7 Клас

Expanding the Square: A Comprehensive Guide to (7x-2y)^2

In algebra, expanding squares is a fundamental concept that helps us simplify complex expressions and solve equations. In this article, we will delve into the world of squares and explore the process of expanding the square of a binomial expression, specifically (7x-2y)^2. We will break down the steps involved, provide examples, and offer tips and tricks to make this process easier to understand and apply.

What is a Binomial Square?

A binomial square is a special type of algebraic expression that can be written in the form (a+b)^2 or (a-b)^2, where 'a' and 'b' are variables or constants. When we expand a binomial square, we get a quadratic expression that can be simplified to a single term. In the case of (7x-2y)^2, we have a binomial square with two terms: 7x and -2y.

Expanding the Square

To expand the square of (7x-2y)^2, we will use the formula:

(a-b)^2 = a^2 - 2ab + b^2

In this case, a = 7x and b = -2y. Plugging these values into the formula, we get:

(7x-2y)^2 = (7x)^2 - 2(7x)(-2y) + (-2y)^2

Step 1: Square the First Term

The first step is to square the first term, 7x. To do this, we multiply 7x by itself:

(7x)^2 = 49x^2

Step 2: Multiply the Terms

Next, we multiply the first term, 7x, by the second term, -2y. To do this, we multiply 7x by -2y:

2(7x)(-2y) = -28xy

Step 3: Square the Second Term

Finally, we square the second term, -2y. To do this, we multiply -2y by itself:

(-2y)^2 = 4y^2

Putting it All Together

Now that we have squared the first term, multiplied the terms, and squared the second term, we can put it all together to get the expanded form of (7x-2y)^2:

(7x-2y)^2 = 49x^2 - 28xy + 4y^2

Tips and Tricks

• When expanding a binomial square, always start by squaring the first term.
• When multiplying the terms, remember to multiply the coefficients (numbers in front of the variables) and the variables separately.
• When squaring the second term, remember to multiply the coefficient by itself and the variable by itself.

Examples

• Expand the square of (3x+2y)^2.
• Expand the square of (2x-3y)^2.

Solution 1: (3x+2y)^2

Using the formula (a+b)^2 = a^2 + 2ab + b^2, we get:

(3x+2y)^2 = (3x)^2 + 2(3x)(2y) + (2y)^2 = 9x^2 + 12xy + 4y^2

Solution 2: (2x-3y)^2

Using the formula (a-b)^2 = a^2 - 2ab + b^2, we get:

(2x-3y)^2 = (2x)^2 - 2(2x)(3y) + (-3y)^2 = 4x^2 - 12xy + 9y^2

Expanding the square of a binomial expression is a fundamental concept in algebra that helps us simplify complex expressions and solve equations. In this article, we have explored the process of expanding the square of (7x-2y)^2, broken down the steps involved, and provided examples to make this process easier to understand and apply. By following the tips and tricks outlined in this article, you will be able to expand binomial squares with confidence and accuracy.
Frequently Asked Questions: Expanding the Square of a Binomial Expression

Q: What is a binomial square?

A: A binomial square is a special type of algebraic expression that can be written in the form (a+b)^2 or (a-b)^2, where 'a' and 'b' are variables or constants.

Q: How do I expand a binomial square?

A: To expand a binomial square, you can use the formula:

(a-b)^2 = a^2 - 2ab + b^2

or

(a+b)^2 = a^2 + 2ab + b^2

Q: What is the difference between (a+b)^2 and (a-b)^2?

A: The main difference between (a+b)^2 and (a-b)^2 is the sign of the middle term. In (a+b)^2, the middle term is 2ab, while in (a-b)^2, the middle term is -2ab.

Q: How do I square the first term?

A: To square the first term, you multiply it by itself. For example, if the first term is 3x, you would square it by multiplying 3x by 3x, which gives you 9x^2.

Q: How do I multiply the terms?

A: To multiply the terms, you multiply the coefficients (numbers in front of the variables) and the variables separately. For example, if you have 2x and 3y, you would multiply 2x by 3y to get 6xy.

Q: How do I square the second term?

A: To square the second term, you multiply it by itself. For example, if the second term is 2y, you would square it by multiplying 2y by 2y, which gives you 4y^2.

Q: What if I have a binomial square with more than two terms?

A: If you have a binomial square with more than two terms, you can still use the formula (a+b)^2 or (a-b)^2, but you will need to expand each term separately. For example, if you have (2x+3y+4z)^2, you would expand each term separately and then combine the results.

Q: Can I use a calculator to expand a binomial square?

A: Yes, you can use a calculator to expand a binomial square. However, it's always a good idea to double-check your work by hand to make sure you get the correct answer.

Q: What if I make a mistake when expanding a binomial square?

A: If you make a mistake when expanding a binomial square, don't worry! Just go back and recheck your work. You can also try using a different method, such as using a formula or a calculator, to get the correct answer.

Q: Are there any special cases I should know about when expanding a binomial square?

A: Yes, there are a few special cases you should know about when expanding a binomial square. For example, if you have a binomial square with a zero coefficient, you can ignore that term. Additionally, if you have a binomial square with a negative coefficient, you will need to take the negative sign into account when expanding the square.

Q: Can I use the formula (a+b)^2 or (a-b)^2 to expand a binomial square with variables on both sides?

A: Yes, you can use the formula (a+b)^2 or (a-b)^2 to expand a binomial square with variables on both sides. However, you will need to be careful when multiplying the terms to make sure you get the correct answer.

Q: Are there any other formulas I can use to expand a binomial square?

A: Yes, there are several other formulas you can use to expand a binomial square, including:

• (a+b)^2 = a^2 + 2ab + b^2
• (a-b)^2 = a^2 - 2ab + b^2
• (a+b+c)^2 = a^2 + 2ab + 2ac + b^2 + 2bc + c^2
• (a-b+c)^2 = a^2 - 2ab + 2ac - b^2 + 2bc - c^2

These formulas can be useful when expanding binomial squares with multiple terms.