Вычислите (49^2 - 38 * 49 + 19^2) / (25^2 - 5^2) В Виде Десятичной Дроби

by Scholario Team 73 views

Hey guys! Today, we're diving into a fun little math problem that involves simplifying a complex expression. We're going to break down each part step by step, so don't worry if it looks intimidating at first. Our mission is to calculate the value of the following expression:

4923849+19225252\frac{49^2 - 38 \cdot 49 + 19^2}{25^2 - 5^2}

And, just to make things interesting, we need to present our answer as a decimal. So, let's roll up our sleeves and get started!

Understanding the Numerator: 49^2 - 38 * 49 + 19^2

Okay, let's first focus on the numerator, which is the top part of our fraction: 49^2 - 38 * 49 + 19^2. This might look a bit tricky, but there's a clever way to simplify it. Notice anything familiar? It might remind you of a perfect square trinomial, which has the form a^2 - 2ab + b^2. If we can somehow fit our expression into this form, we can simplify it quite easily.

Let’s rewrite the middle term, -38 * 49, to see if we can make this connection clearer. We can express 38 as 2 * 19, so our middle term becomes -2 * 19 * 49. Now our numerator looks like this:

49221949+19249^2 - 2 \cdot 19 \cdot 49 + 19^2

Ah-ha! Do you see it now? It perfectly matches the form a^2 - 2ab + b^2, where a = 49 and b = 19. This is fantastic because we know that:

a22ab+b2=(ab)2a^2 - 2ab + b^2 = (a - b)^2

So, we can rewrite our numerator as:

(4919)2(49 - 19)^2

Which simplifies to:

302=90030^2 = 900

Awesome! We've conquered the numerator. Now, let's move on to the denominator.

Simplifying the Denominator: 25^2 - 5^2

Now, let's tackle the denominator: 25^2 - 5^2. This part looks like another familiar pattern, the difference of squares. Remember this handy formula?

a2b2=(a+b)(ab)a^2 - b^2 = (a + b)(a - b)

In our case, a = 25 and b = 5. So, we can rewrite our denominator as:

(25+5)(255)(25 + 5)(25 - 5)

This simplifies to:

(30)(20)=600(30)(20) = 600

Fantastic! We've simplified the denominator too. Now we're ready to put it all together.

Putting It All Together: 900 / 600

Okay, we've simplified the numerator to 900 and the denominator to 600. Our original expression now looks like this:

900600\frac{900}{600}

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 300:

900÷300600÷300=32\frac{900 \div 300}{600 \div 300} = \frac{3}{2}

Now, to present our answer as a decimal, we simply divide 3 by 2:

32=1.5\frac{3}{2} = 1.5

Final Answer

And there you have it! The value of the expression

4923849+19225252\frac{49^2 - 38 \cdot 49 + 19^2}{25^2 - 5^2}

presented as a decimal is 1.5. Great job, guys! We took a seemingly complex problem and broke it down into manageable steps. Remember, recognizing patterns and using algebraic identities can make these calculations much easier. Keep practicing, and you'll become math superstars in no time!


Mastering Algebraic Expressions: A Comprehensive Guide

Algebraic expressions can sometimes seem daunting, but with the right approach and understanding of fundamental concepts, they become much more manageable. In this comprehensive guide, we'll delve deeper into the world of algebraic expressions, covering essential techniques and strategies to tackle even the most challenging problems. Let's embark on this mathematical journey together and unlock the power of algebraic manipulation!

Understanding Key Concepts

Before diving into specific techniques, it's crucial to grasp some key concepts that form the foundation of algebraic expressions.

  • Variables: Variables are symbols (usually letters) that represent unknown quantities. For example, in the expression 3x + 5, 'x' is a variable.
  • Constants: Constants are fixed numerical values that don't change. In the same expression, 3 and 5 are constants.
  • Coefficients: A coefficient is a number that multiplies a variable. In 3x + 5, 3 is the coefficient of 'x'.
  • Terms: Terms are the individual components of an expression, separated by addition or subtraction. In 3x + 5, '3x' and '5' are terms.
  • Operators: Operators are symbols that perform mathematical operations, such as addition (+), subtraction (-), multiplication (*), and division (/).

Techniques for Simplifying Algebraic Expressions

Simplifying algebraic expressions involves rewriting them in a more concise and manageable form. Here are some key techniques:

  1. Combining Like Terms:

    Like terms are terms that have the same variable raised to the same power. To combine like terms, simply add or subtract their coefficients. For example:

    3x + 5x - 2x = (3 + 5 - 2)x = 6x

    4y^2 - 2y^2 + y^2 = (4 - 2 + 1)y^2 = 3y^2

  2. Distributive Property:

    The distributive property allows you to multiply a factor across terms within parentheses. The rule is:

    a(b + c) = ab + ac

    For example:

    2(x + 3) = 2x + 6

    -3(2y - 1) = -6y + 3

  3. Factoring:

    Factoring is the reverse process of the distributive property. It involves breaking down an expression into its factors. For example:

    4x + 8 = 4(x + 2)

    3y^2 - 6y = 3y(y - 2)

  4. Using Algebraic Identities:

    Algebraic identities are equations that are always true, regardless of the values of the variables. They provide shortcuts for simplifying certain types of expressions. Some common identities include:

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

    Let's see how these identities can be applied.

Applying Algebraic Identities: A Detailed Look

Algebraic identities are powerful tools that can significantly simplify complex expressions. Let's explore how to use them effectively.

  1. (a + b)^2 = a^2 + 2ab + b^2:

    This identity is the square of a binomial sum. For example, to expand (x + 4)^2, we can use this identity:

    (x + 4)^2 = x^2 + 2(x)(4) + 4^2 = x^2 + 8x + 16

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

    This identity is the square of a binomial difference. For example, to expand (2y - 3)^2, we use this identity:

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

  3. (a + b)(a - b) = a^2 - b^2:

    This identity is the difference of squares. For example, to simplify (3z + 2)(3z - 2), we use this identity:

    (3z + 2)(3z - 2) = (3z)^2 - 2^2 = 9z^2 - 4

Example Problems and Solutions

Let's work through some example problems to solidify our understanding.

Problem 1: Simplify the expression (2x + 3)(x - 2) + 5x

Solution:

  1. First, expand the product using the distributive property (also known as the FOIL method):

    (2x + 3)(x - 2) = 2x(x) + 2x(-2) + 3(x) + 3(-2) = 2x^2 - 4x + 3x - 6

  2. Combine like terms:

    2x^2 - 4x + 3x - 6 = 2x^2 - x - 6

  3. Add the remaining term:

    2x^2 - x - 6 + 5x = 2x^2 + 4x - 6

    Therefore, the simplified expression is 2x^2 + 4x - 6.

Problem 2: Factor the expression 9y^2 - 16

Solution:

  1. Recognize that this is a difference of squares, where a^2 = 9y^2 and b^2 = 16.

  2. Find 'a' and 'b' by taking the square root:

    a = 3y and b = 4

  3. Apply the difference of squares identity:

    9y^2 - 16 = (3y + 4)(3y - 4)

    Thus, the factored expression is (3y + 4)(3y - 4).

Problem 3: Expand and simplify (a - 2b)^2 - (a + b)(a - b)

Solution:

  1. Expand the square using the binomial square identity:

    (a - 2b)^2 = a^2 - 2(a)(2b) + (2b)^2 = a^2 - 4ab + 4b^2

  2. Expand the product using the difference of squares identity:

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

  3. Subtract the second expression from the first:

    (a^2 - 4ab + 4b^2) - (a^2 - b^2) = a^2 - 4ab + 4b^2 - a^2 + b^2

  4. Combine like terms:

    a^2 - 4ab + 4b^2 - a^2 + b^2 = -4ab + 5b^2

    Hence, the simplified expression is -4ab + 5b^2.

Tips and Tricks for Success

  • Practice Regularly: The key to mastering algebraic expressions is consistent practice. Work through a variety of problems to build your skills and confidence.
  • Break Down Complex Problems: If an expression looks intimidating, break it down into smaller, more manageable steps.
  • Recognize Patterns: Familiarize yourself with common algebraic patterns and identities. This will help you simplify expressions more efficiently.
  • Double-Check Your Work: Always double-check your work to avoid errors, especially when dealing with multiple steps.
  • Seek Help When Needed: Don't hesitate to ask for help from teachers, tutors, or online resources if you're struggling with a particular concept.

Conclusion

Algebraic expressions are a fundamental part of mathematics, and mastering them opens doors to more advanced concepts. By understanding key concepts, practicing simplification techniques, and leveraging algebraic identities, you can confidently tackle even the most challenging expressions. Remember, the journey to algebraic mastery is a marathon, not a sprint. Keep practicing, stay curious, and you'll be amazed at how far you can go!


Tackling Complex Algebraic Problems: Advanced Techniques

Alright, math enthusiasts! Now that we've covered the basics and some intermediate techniques for simplifying algebraic expressions, let's crank things up a notch. We're going to dive into some advanced techniques that will help you tackle those really challenging problems. Get ready to level up your algebra game!

More on Factoring: Beyond the Basics

We touched on factoring earlier, but there's so much more to explore. Factoring is an incredibly powerful tool for simplifying expressions, solving equations, and understanding the structure of polynomials. Let's delve into some advanced factoring techniques.

  1. Factoring Trinomials with a Leading Coefficient:

    When you have a trinomial of the form ax^2 + bx + c, where a is not equal to 1, the factoring process becomes a bit more involved. One common method is the