Solving For X In The Equation 4(4x + 3)(9x + 10) = 3x

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Introduction

In this article, we will delve into the mathematical problem of finding the value of x in the equation 4(4x + 3)(9x + 10), where it is stated that this expression represents the triplicate value of x. This problem combines algebraic manipulation with a touch of number theory, requiring a careful step-by-step approach to arrive at the solution. We will explore the concepts involved, break down the equation, and solve for x using algebraic techniques. Understanding this problem not only enhances one's problem-solving skills but also deepens the comprehension of how algebraic expressions can model real-world scenarios.

Understanding the Problem

To effectively tackle this equation, it's crucial to first understand the terms and concepts involved. The expression 4(4x + 3)(9x + 10) is an algebraic expression that involves multiplication and addition. The term "triplicate value of x" refers to three times the value of x, which can be mathematically represented as 3x. Thus, the core of the problem lies in equating the given expression to 3x and solving for x. This involves expanding the expression, simplifying the equation, and using algebraic methods to isolate x on one side of the equation. Understanding these basics is the first step towards successfully solving the problem.

Setting Up the Equation

The first step in solving this problem is to set up the equation correctly. We are given that 4(4x + 3)(9x + 10) is equal to the triplicate value of x, which is 3x. Therefore, the equation can be written as:

4(4x + 3)(9x + 10) = 3x

This equation forms the basis for our solution. It is a polynomial equation, and solving it will involve expanding the terms, simplifying the expression, and then finding the roots of the polynomial. Setting up the equation correctly is crucial because any error at this stage will propagate through the rest of the solution. Now that we have the equation, we can move on to the next step, which involves expanding the terms and simplifying the equation.

Expanding and Simplifying the Equation

Expanding the Expression

To solve the equation 4(4x + 3)(9x + 10) = 3x, we first need to expand the expression on the left side. This involves multiplying the binomials (4x + 3) and (9x + 10) and then multiplying the result by 4. The expansion process is as follows:

  1. Multiply (4x + 3) by (9x + 10): (4x + 3)(9x + 10) = 4x * 9x + 4x * 10 + 3 * 9x + 3 * 10 = 36x² + 40x + 27x + 30 = 36x² + 67x + 30
  2. Multiply the result by 4: 4(36x² + 67x + 30) = 144x² + 268x + 120

So, the expanded form of the left side of the equation is 144x² + 268x + 120. This expansion is a critical step in simplifying the equation and bringing it closer to a solvable form.

Simplifying the Equation

Now that we have expanded the expression, the equation becomes:

144x² + 268x + 120 = 3x

To simplify this equation, we need to move all the terms to one side, setting the equation equal to zero. This will give us a quadratic equation in the standard form (ax² + bx + c = 0). Subtracting 3x from both sides, we get:

144x² + 268x - 3x + 120 = 0

Combining like terms, the equation simplifies to:

144x² + 265x + 120 = 0

This is a quadratic equation in the standard form, where a = 144, b = 265, and c = 120. Now that we have a simplified quadratic equation, we can proceed to solve for x using various methods such as factoring, completing the square, or the quadratic formula.

Solving the Quadratic Equation

Methods to Solve Quadratic Equations

To solve the quadratic equation 144x² + 265x + 120 = 0, we have several methods at our disposal. The most common methods include:

  1. Factoring: This method involves breaking down the quadratic expression into two binomial factors. If the equation can be factored, it provides a straightforward way to find the roots.

  2. Completing the Square: This method involves manipulating the equation to form a perfect square trinomial on one side, which can then be easily solved.

  3. Quadratic Formula: This is a general formula that can be used to solve any quadratic equation. The formula is:

    x = (-b ± √(b² - 4ac)) / (2a)

    where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.

Given the coefficients of our equation (a = 144, b = 265, c = 120), factoring might be challenging. Completing the square is also a viable method, but the quadratic formula is often the most direct approach for equations that are not easily factored. Therefore, we will use the quadratic formula to solve for x.

Applying the Quadratic Formula

Using the quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), and the coefficients from our equation 144x² + 265x + 120 = 0 (a = 144, b = 265, c = 120), we can substitute these values into the formula:

x = (-265 ± √(265² - 4 * 144 * 120)) / (2 * 144)

First, let's calculate the discriminant (b² - 4ac):

Discriminant = 265² - 4 * 144 * 120 = 70225 - 69120 = 1105

Now, substitute the discriminant back into the quadratic formula:

x = (-265 ± √1105) / (288)

This gives us two possible values for x:

x₁ = (-265 + √1105) / 288 x₂ = (-265 - √1105) / 288

Calculating the Values of x

To find the approximate values of x, we need to calculate the square root of 1105 and then perform the arithmetic operations. The square root of 1105 is approximately 33.24.

Now, we can calculate the two values of x:

x₁ = (-265 + 33.24) / 288 = -231.76 / 288 ≈ -0.805

x₂ = (-265 - 33.24) / 288 = -298.24 / 288 ≈ -1.036

So, the two possible values for x are approximately -0.805 and -1.036. These values represent the solutions to the quadratic equation and, consequently, the solutions to the original problem.

Verifying the Solutions

Substituting the Values Back into the Equation

To ensure the accuracy of our solutions, we must verify them by substituting the values of x we found back into the original equation: 4(4x + 3)(9x + 10) = 3x. This step is crucial in confirming that the solutions are correct and that no errors were made during the algebraic manipulation.

We have two values for x: x₁ ≈ -0.805 and x₂ ≈ -1.036. We will substitute each value into the original equation and check if the equation holds true.

Verification Process

  1. Verification for x₁ ≈ -0.805:

    Substitute x = -0.805 into the equation:

    4(4(-0.805) + 3)(9(-0.805) + 10) = 3(-0.805)

    First, calculate the values inside the parentheses:

    4(-3.22 + 3)(-7.245 + 10) = -2.415 4(-0.22)(2.755) = -2.415 4(-0.6061) ≈ -2.415 -2.4244 ≈ -2.415

    The left side is approximately equal to the right side, so x₁ ≈ -0.805 is a valid solution.

  2. Verification for x₂ ≈ -1.036:

    Substitute x = -1.036 into the equation:

    4(4(-1.036) + 3)(9(-1.036) + 10) = 3(-1.036)

    First, calculate the values inside the parentheses:

    4(-4.144 + 3)(-9.324 + 10) = -3.108 4(-1.144)(0.676) = -3.108 4(-0.773424) ≈ -3.108 -3.093696 ≈ -3.108

    The left side is approximately equal to the right side, so x₂ ≈ -1.036 is also a valid solution.

Conclusion of Verification

Both values of x we found, x₁ ≈ -0.805 and x₂ ≈ -1.036, satisfy the original equation. This confirms that our solutions are accurate and that the process we followed to solve the equation was correct. Verification is a crucial step in mathematical problem-solving, as it helps to ensure that the solutions obtained are valid and that no errors were made during the calculations.

Conclusion

In conclusion, we have successfully found the values of x in the equation 4(4x + 3)(9x + 10) = 3x. By expanding and simplifying the equation, we transformed it into a quadratic equation, which we then solved using the quadratic formula. We obtained two solutions: x₁ ≈ -0.805 and x₂ ≈ -1.036. These solutions were verified by substituting them back into the original equation, confirming their validity.

This problem demonstrates the importance of understanding algebraic concepts, such as expanding expressions, simplifying equations, and using the quadratic formula. It also highlights the significance of verification in ensuring the accuracy of solutions. The step-by-step approach we followed can be applied to similar mathematical problems, enhancing problem-solving skills and deepening mathematical understanding.

Summary of Key Steps

  1. Understanding the Problem: We began by understanding the given equation and the concept of the triplicate value of x.
  2. Setting Up the Equation: We correctly set up the equation 4(4x + 3)(9x + 10) = 3x.
  3. Expanding the Expression: We expanded the expression on the left side of the equation.
  4. Simplifying the Equation: We simplified the equation into a standard quadratic form.
  5. Solving the Quadratic Equation: We used the quadratic formula to find the values of x.
  6. Verifying the Solutions: We verified the solutions by substituting them back into the original equation.
  7. Conclusion: We summarized the process and the solutions obtained.

This comprehensive approach ensures that we not only find the solutions but also understand the underlying concepts and techniques involved in solving the problem.