Find The Value Of C In The Equation -2x^3(cx^3 + X^2) = 10x^6 - 2x^5

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In the realm of algebra, solving equations and determining the values of unknown variables is a fundamental skill. This article delves into a specific algebraic problem where we aim to find the value of the constant c that satisfies a given equation. The equation in question is: −2x3(cx3+x2)=10x6−2x5-2x^3(cx^3 + x^2) = 10x^6 - 2x^5. This problem combines polynomial multiplication, simplification, and equation solving techniques, making it an excellent exercise for students and enthusiasts alike. Our goal is to provide a step-by-step solution, ensuring clarity and understanding for readers of all backgrounds. Let's embark on this mathematical journey and unravel the value of c that makes the equation ring true.

Understanding the Problem

Before diving into the solution, it's crucial to understand the problem thoroughly. We are given an equation that involves a variable x and a constant c. The equation is: −2x3(cx3+x2)=10x6−2x5-2x^3(cx^3 + x^2) = 10x^6 - 2x^5. Our task is to find the specific value of c that makes this equation a true statement for all values of x. This means that when we substitute the correct value of c into the equation, the left-hand side (LHS) should be identical to the right-hand side (RHS). To achieve this, we will employ algebraic manipulation techniques, including the distributive property and simplification of polynomial expressions. Understanding the nature of the problem sets the stage for a systematic approach to finding the solution.

Step-by-Step Solution

Now, let's proceed with a detailed, step-by-step solution to determine the value of c. Each step will be explained to ensure clarity and understanding.

1. Distribute the Term

The first step involves applying the distributive property to the left-hand side (LHS) of the equation. The distributive property states that a(b + c) = ab + ac. In our case, we need to distribute −2x3-2x^3 across the terms inside the parenthesis (cx3+x2)(cx^3 + x^2).

So, −2x3(cx3+x2)-2x^3(cx^3 + x^2) becomes (−2x3∗cx3)+(−2x3∗x2)(-2x^3 * cx^3) + (-2x^3 * x^2).

2. Simplify the Terms

Next, we simplify each term obtained after distribution. Remember the rule of exponents: xm∗xn=x(m+n)x^m * x^n = x^(m+n).

The first term, −2x3∗cx3-2x^3 * cx^3, simplifies to −2cx(3+3)=−2cx6-2cx^(3+3) = -2cx^6.

The second term, −2x3∗x2-2x^3 * x^2, simplifies to −2x(3+2)=−2x5-2x^(3+2) = -2x^5.

Therefore, the LHS of the equation now becomes −2cx6−2x5-2cx^6 - 2x^5.

3. Rewrite the Equation

Now, let's rewrite the entire equation with the simplified LHS:

−2cx6−2x5=10x6−2x5-2cx^6 - 2x^5 = 10x^6 - 2x^5

4. Isolate the Term with cc

Our goal is to find the value of c. To do this, we need to isolate the term that contains c. Notice that both sides of the equation have a term −2x5-2x^5. We can eliminate this term by adding 2x52x^5 to both sides of the equation:

−2cx6−2x5+2x5=10x6−2x5+2x5-2cx^6 - 2x^5 + 2x^5 = 10x^6 - 2x^5 + 2x^5

This simplifies to:

−2cx6=10x6-2cx^6 = 10x^6

5. Solve for cc

Now, we have a simpler equation with only one term containing c. To solve for c, we need to divide both sides of the equation by −2x6-2x^6:

(−2cx6)/(−2x6)=(10x6)/(−2x6)(-2cx^6) / (-2x^6) = (10x^6) / (-2x^6)

This simplifies to:

c=−5c = -5

6. Verify the Solution

To ensure our solution is correct, we need to substitute the value of c back into the original equation and check if the LHS equals the RHS.

Original equation: −2x3(cx3+x2)=10x6−2x5-2x^3(cx^3 + x^2) = 10x^6 - 2x^5

Substitute c=−5c = -5:

−2x3(−5x3+x2)=10x6−2x5-2x^3(-5x^3 + x^2) = 10x^6 - 2x^5

Distribute −2x3-2x^3:

(−2x3∗−5x3)+(−2x3∗x2)=10x6−2x5(-2x^3 * -5x^3) + (-2x^3 * x^2) = 10x^6 - 2x^5

Simplify:

10x6−2x5=10x6−2x510x^6 - 2x^5 = 10x^6 - 2x^5

Since the LHS equals the RHS, our solution is correct.

Final Answer

Therefore, the value of c that makes the statement true is -5. This comprehensive step-by-step solution ensures a clear understanding of the process, from distributing terms to isolating the variable and verifying the solution.

Common Mistakes to Avoid

While solving algebraic equations, it's easy to make mistakes if one isn't careful. Here are some common pitfalls to avoid when tackling problems like this:

  • Incorrect Distribution: A frequent error is misapplying the distributive property. Ensure that you multiply the term outside the parenthesis with each term inside. For example, in our case, −2x3-2x^3 must be multiplied with both cx3cx^3 and x2x^2.
  • Sign Errors: Pay close attention to signs, especially when dealing with negative numbers. A simple sign error can lead to an incorrect answer. For instance, multiplying −2x3-2x^3 by −5x3-5x^3 should result in a positive 10x610x^6.
  • Exponent Rules: Mistakes in applying exponent rules are common. Remember that when multiplying terms with the same base, you add the exponents (xm∗xn=x(m+n)x^m * x^n = x^(m+n)). A common error is to multiply the exponents instead of adding them.
  • Incorrect Simplification: Ensure you simplify the equation correctly at each step. This includes combining like terms and performing arithmetic operations accurately.
  • Forgetting to Verify: Always verify your solution by substituting it back into the original equation. This helps catch any errors made during the solving process.
  • Rushing Through Steps: Algebra requires attention to detail. Rushing through steps can lead to mistakes. Take your time, write each step clearly, and double-check your work.

By being mindful of these common mistakes, you can significantly improve your accuracy and confidence in solving algebraic equations.

Practice Problems

To reinforce your understanding of finding the value of constants in algebraic equations, here are some practice problems:

  1. Find the value of k in the equation: 3x2(kx4−2x)=12x6−6x33x^2(kx^4 - 2x) = 12x^6 - 6x^3
  2. Determine the value of m in the equation: −4x5(mx2+3x3)=−20x7−12x8-4x^5(mx^2 + 3x^3) = -20x^7 - 12x^8
  3. Solve for p in the equation: 2x4(px3−x5)=14x7−2x92x^4(px^3 - x^5) = 14x^7 - 2x^9
  4. What value of q makes the statement true: −x2(qx5+4x2)=−9x7−4x4-x^2(qx^5 + 4x^2) = -9x^7 - 4x^4
  5. Find the constant r in the equation: 5x3(rx2−x4)=35x5−5x75x^3(rx^2 - x^4) = 35x^5 - 5x^7

Working through these practice problems will help solidify your skills and understanding of the concepts discussed in this article. Remember to follow the step-by-step approach outlined earlier, and don't forget to verify your solutions.

Conclusion

In conclusion, finding the value of a constant that makes an algebraic statement true involves a systematic approach that combines algebraic manipulation techniques. In this article, we successfully determined that the value of c in the equation −2x3(cx3+x2)=10x6−2x5-2x^3(cx^3 + x^2) = 10x^6 - 2x^5 is -5. We achieved this by carefully applying the distributive property, simplifying terms, isolating the variable, and verifying our solution. Furthermore, we highlighted common mistakes to avoid and provided practice problems to reinforce understanding. By mastering these techniques, you can confidently tackle a wide range of algebraic problems and enhance your mathematical prowess. Algebra is a foundational pillar of mathematics, and the ability to solve equations is a valuable skill that extends far beyond the classroom. Embrace the challenges, practice diligently, and watch your algebraic abilities flourish.

What is the value of cc that satisfies the equation −2x3(cx3+x2)=10x6−2x5-2x^3(cx^3 + x^2) = 10x^6 - 2x^5?

Find the Value of c in the Equation -2x3(cx3 + x^2) = 10x^6 - 2x^5