Quickly Solve: 2(3x−4)² − 3(2x−5)(2x+5) Calculation

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Quickly Solve: 2(3x−4)² − 3(2x−5)(2x+5) Calculation

Alright, guys, let's break down this math problem step by step. We need to calculate 2(3x−4)² − 3(2x−5)(2x+5) − 4(x+3)² − (2x−3)(3x+2) using short calculation formulas. Buckle up, it's gonna be a ride!

Step 1: Expand the Terms

First, we'll expand each term using the formulas for the square of a binomial and the difference of squares. This involves careful application of algebraic identities to simplify each expression before combining them.

Expanding 2(3x−4)²

So, we're tackling 2(3x - 4)² first. Remember the formula (a - b)² = a² - 2ab + b². Applying this, we get:

2 * ( (3x)² - 2 * (3x) * 4 + 4² )

2 * ( 9x² - 24x + 16 )

Distribute the 2:

18x² - 48x + 32

Expansion Insights: Expanding this term involves squaring a binomial and then distributing a constant. Pay close attention to the signs and coefficients to avoid errors. This is a foundational step that sets the stage for simplifying the entire expression. The key here is correctly applying the binomial square formula and ensuring that all terms are accurately multiplied.

Expanding −3(2x−5)(2x+5)

Now, let's expand −3(2x - 5)(2x + 5). We'll use the difference of squares formula: (a - b)(a + b) = a² - b².

-3 * ( (2x)² - 5² )

-3 * ( 4x² - 25 )

Distribute the -3:

-12x² + 75

Difference of Squares Mastery: Recognizing and applying the difference of squares formula is crucial for simplifying expressions like these quickly. This formula allows us to bypass the traditional FOIL method, saving time and reducing the chance of errors. Always double-check the signs to ensure accurate expansion. Applying the difference of squares correctly is very important here, as it directly impacts the subsequent distribution and simplification steps.

Expanding −4(x+3)²

Next up, we have −4(x + 3)². Again, we use the formula (a + b)² = a² + 2ab + b².

-4 * ( x² + 2 * x * 3 + 3² )

-4 * ( x² + 6x + 9 )

Distribute the -4:

-4x² - 24x - 36

Binomial Square Caution: Expanding this term requires careful attention to the signs, especially since we are multiplying by a negative number. Ensuring that each term is correctly multiplied by -4 is essential for maintaining accuracy throughout the simplification process. Make sure to distribute the -4 correctly to each term inside the parenthesis to avoid mistakes.

Expanding −(2x−3)(3x+2)

Finally, let's expand −(2x - 3)(3x + 2). We'll use the FOIL (First, Outer, Inner, Last) method.

- * ( 2x * 3x + 2x * 2 - 3 * 3x - 3 * 2 )

- * ( 6x² + 4x - 9x - 6 )

- * ( 6x² - 5x - 6 )

Distribute the negative sign:

-6x² + 5x + 6

FOIL Method Precision: Applying the FOIL method accurately involves multiplying each term in the first binomial by each term in the second binomial and then combining like terms. The negative sign in front of the parentheses requires distributing it to each term after the expansion. This step is crucial for obtaining the correct signs in the final expression. Remember to distribute that negative sign to every term inside the parenthesis.

Step 2: Combine Like Terms

Now that we've expanded all the terms, let's combine the like terms:

(18x² - 48x + 32) + (-12x² + 75) + (-4x² - 24x - 36) + (-6x² + 5x + 6)

Combine the x² terms:

18x² - 12x² - 4x² - 6x² = (18 - 12 - 4 - 6)x² = -4x²

Combine the x terms:

-48x - 24x + 5x = (-48 - 24 + 5)x = -67x

Combine the constant terms:

32 + 75 - 36 + 6 = 77

Step 3: Final Result

Putting it all together, we get:

-4x² - 67x + 77

Consolidating Like Terms: Combining like terms is the final step in simplifying the expanded expression. It involves identifying and adding or subtracting terms with the same variable and exponent. This step requires careful attention to the signs and coefficients to ensure accuracy. Always double-check that you've combined like terms correctly.

So, the final answer is:

-4x² - 67x + 77

There you have it! We've successfully calculated 2(3x−4)² − 3(2x−5)(2x+5) − 4(x+3)² − (2x−3)(3x+2) using short calculation formulas. Remember to take it one step at a time, and you'll be a math whiz in no time!

Key Concepts Used

Square of a Binomial

The formula for the square of a binomial is a fundamental algebraic identity that helps simplify expressions quickly. It comes in two forms:

  • (a + b)² = a² + 2ab + b²
  • (a - b)² = a² - 2ab + b²

This formula is used to expand expressions where a binomial is raised to the power of 2. Instead of multiplying (a + b)(a + b) manually, you can directly apply the formula to get the expanded form. This not only saves time but also reduces the chances of making errors.

For example, when expanding (x + 3)², we identify a as x and b as 3. Plugging these values into the formula, we get:

(x + 3)² = x² + 2(x)(3) + 3² = x² + 6x + 9

Understanding and memorizing this formula is crucial for efficiently simplifying algebraic expressions. Practice applying it in various scenarios to become proficient.

Difference of Squares

The difference of squares is another essential algebraic identity that simplifies the product of two binomials. The formula is:

(a - b)(a + b) = a² - b²

This formula states that when you multiply two binomials that are identical except for the sign between their terms, the result is the square of the first term minus the square of the second term. Recognizing this pattern can significantly speed up calculations.

For example, when expanding (2x - 5)(2x + 5), we identify a as 2x and b as 5. Applying the formula, we get:

(2x - 5)(2x + 5) = (2x)² - 5² = 4x² - 25

The difference of squares formula is particularly useful in simplifying expressions that appear in various mathematical contexts, including factoring and solving equations.

FOIL Method

The FOIL method is a technique used to multiply two binomials. FOIL stands for First, Outer, Inner, Last, indicating the order in which terms should be multiplied:

  • First: Multiply the first terms in each binomial.
  • Outer: Multiply the outer terms in each binomial.
  • Inner: Multiply the inner terms in each binomial.
  • Last: Multiply the last terms in each binomial.

After multiplying the terms, combine like terms to simplify the expression.

For example, when expanding (2x - 3)(3x + 2), we apply the FOIL method as follows:

  • First: 2x * 3x = 6x²
  • Outer: 2x * 2 = 4x
  • Inner: -3 * 3x = -9x
  • Last: -3 * 2 = -6

Combining these terms, we get:

6x² + 4x - 9x - 6 = 6x² - 5x - 6

The FOIL method is a systematic way to ensure that all terms in the binomials are multiplied correctly, making it a valuable tool for algebraic manipulations.

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply a single term by multiple terms within a set of parentheses. The property is expressed as:

a(b + c) = ab + ac

This means that you multiply the term outside the parentheses (a) by each term inside the parentheses (b and c). The distributive property is essential for expanding and simplifying algebraic expressions.

For example, when expanding -4(x + 3)², after applying the square of a binomial, we have:

-4(x² + 6x + 9)

Now, we distribute the -4 to each term inside the parentheses:

-4 * x² + (-4) * 6x + (-4) * 9 = -4x² - 24x - 36

The distributive property is used extensively in algebra and is crucial for correctly expanding and simplifying expressions.

Understanding and mastering these key concepts—square of a binomial, difference of squares, FOIL method, and distributive property—will greatly enhance your ability to simplify algebraic expressions quickly and accurately. Keep practicing, and you'll become more confident in your algebraic skills!