Solving 3x² + 13x - 7 = 0 A Step-by-Step Guide To Two Decimal Places

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Solving quadratic equations is a fundamental skill in algebra, and the equation 3x² + 13x - 7 = 0 is a classic example. This article will provide a detailed, step-by-step guide on how to solve this equation, focusing on using the quadratic formula to find the solutions to two decimal places. We'll explore the underlying concepts, the application of the formula, and the interpretation of the results. Whether you're a student grappling with algebra or someone looking to refresh your math skills, this guide will equip you with the knowledge to tackle similar problems.

Understanding Quadratic Equations

To effectively solve the equation 3x² + 13x - 7 = 0, it's crucial to first understand what a quadratic equation is and its general form. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in this case, x) is 2. The general form of a quadratic equation is:

ax² + bx + c = 0

Where a, b, and c are constants, and a is not equal to 0. If a were 0, the equation would become a linear equation instead of a quadratic. In our specific equation, 3x² + 13x - 7 = 0, we can identify the coefficients as follows:

  • a = 3
  • b = 13
  • c = -7

The coefficients a, b, and c play a critical role in determining the solutions (also called roots) of the quadratic equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Factoring involves rewriting the quadratic expression as a product of two linear expressions. This method is efficient when the quadratic expression can be easily factored. Completing the square is another technique that involves manipulating the equation to form a perfect square trinomial, which can then be easily solved. However, the most versatile and universally applicable method is the quadratic formula. The quadratic formula provides a direct way to find the solutions for any quadratic equation, regardless of whether it can be factored easily or not. This makes it an essential tool in algebra.

The Quadratic Formula: A Powerful Tool

The quadratic formula is a cornerstone in solving quadratic equations. It provides a direct method for finding the roots of any quadratic equation in the form ax² + bx + c = 0. The formula is expressed as:

x = (-b ± √(b² - 4ac)) / 2a

This formula may seem complex at first, but it's simply a matter of substituting the values of a, b, and c from the given quadratic equation and simplifying. The ± symbol indicates that there are typically two solutions, one obtained by adding the square root term and the other by subtracting it. These two solutions are the points where the parabola represented by the quadratic equation intersects the x-axis.

The term inside the square root, b² - 4ac, is called the discriminant. The discriminant plays a crucial role in determining the nature of the solutions. There are three possible scenarios:

  1. If b² - 4ac > 0, the equation has two distinct real solutions. This means the parabola intersects the x-axis at two different points.
  2. If b² - 4ac = 0, the equation has exactly one real solution (a repeated root). In this case, the parabola touches the x-axis at only one point.
  3. If b² - 4ac < 0, the equation has two complex solutions (no real solutions). This indicates that the parabola does not intersect the x-axis.

Understanding the discriminant is essential for predicting the type of solutions you'll obtain before even applying the full quadratic formula. This can save time and prevent errors in calculations. In the context of our equation, 3x² + 13x - 7 = 0, we will use the quadratic formula to find the solutions to two decimal places. This requires careful substitution, simplification, and approximation to achieve the desired accuracy.

Applying the Quadratic Formula to 3x² + 13x - 7 = 0

Now, let's apply the quadratic formula to the equation 3x² + 13x - 7 = 0. We've already identified the coefficients:

  • a = 3
  • b = 13
  • c = -7

Substitute these values into the quadratic formula:

x = (-13 ± √(13² - 4 * 3 * -7)) / (2 * 3)

First, we need to simplify the expression under the square root:

13² - 4 * 3 * -7 = 169 + 84 = 253

Now, substitute this back into the formula:

x = (-13 ± √253) / 6

Next, we need to find the square root of 253. Since we need the answer to two decimal places, we'll approximate √253. Using a calculator, we find:

√253 ≈ 15.91

Now, we have:

x = (-13 ± 15.91) / 6

This gives us two solutions:

  1. x₁ = (-13 + 15.91) / 6
  2. x₂ = (-13 - 15.91) / 6

Let's calculate each solution:

  1. x₁ = (2.91) / 6 ≈ 0.49
  2. x₂ = (-28.91) / 6 ≈ -4.82

Therefore, the solutions to the equation 3x² + 13x - 7 = 0, rounded to two decimal places, are approximately 0.49 and -4.82. This step-by-step application of the quadratic formula demonstrates how to systematically solve quadratic equations, ensuring accuracy and clarity in the process.

Step-by-Step Calculation Breakdown

To further clarify the process, let's break down the calculation steps in more detail. This will help ensure a clear understanding of how we arrived at the solutions 0.49 and -4.82. We started with the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

And the equation:

3x² + 13x - 7 = 0

  1. Identify Coefficients: As we mentioned earlier, we identified a = 3, b = 13, and c = -7. Correctly identifying these coefficients is crucial, as any error here will propagate through the rest of the calculation.

  2. Substitute into the Formula: We substituted these values into the quadratic formula:

    x = (-13 ± √(13² - 4 * 3 * -7)) / (2 * 3)

  3. Simplify the Discriminant: The discriminant (b² - 4ac) is the term inside the square root. We calculated it as follows:

    13² - 4 * 3 * -7 = 169 + 84 = 253

    This step is critical because the value of the discriminant determines the nature of the solutions. In this case, since 253 is positive, we know there are two distinct real solutions.

  4. Approximate the Square Root: We found the square root of 253 to two decimal places using a calculator:

    √253 ≈ 15.91

    Approximation is often necessary when the square root is not a whole number, especially when the question requires the answer to a specific number of decimal places.

  5. Substitute the Square Root Value: We substituted the approximate square root value back into the formula:

    x = (-13 ± 15.91) / 6

  6. Calculate the Two Solutions: We then calculated the two solutions separately, one with the plus sign and one with the minus sign:

    • x₁ = (-13 + 15.91) / 6 = 2.91 / 6 ≈ 0.49
    • x₂ = (-13 - 15.91) / 6 = -28.91 / 6 ≈ -4.82

    Each solution represents a point where the quadratic function crosses the x-axis. The positive solution (0.49) is the x-coordinate of the rightmost intersection, and the negative solution (-4.82) is the x-coordinate of the leftmost intersection.

By breaking down the calculation into these steps, we can see how each part of the quadratic formula contributes to the final solutions. This detailed approach helps minimize errors and ensures a clear understanding of the process.

Verifying the Solutions

After finding the solutions to a quadratic equation, it's always a good practice to verify them. This helps to ensure that the calculations were done correctly and that the solutions are accurate. There are a couple of ways to verify the solutions we found for the equation 3x² + 13x - 7 = 0, which were approximately 0.49 and -4.82.

One method is to substitute each solution back into the original equation and check if the equation holds true (i.e., the left side equals zero). Let's start with x₁ ≈ 0.49:

3(0.49)² + 13(0.49) - 7

First, calculate (0.49)²:

  1. 49² ≈ 0.2401

Now, substitute this back into the expression:

3(0.2401) + 13(0.49) - 7 ≈ 0.7203 + 6.37 - 7 ≈ 0.0903

This is close to zero, which is expected given that we rounded the solution to two decimal places. The slight deviation is due to the rounding error.

Next, let's verify x₂ ≈ -4.82:

3(-4.82)² + 13(-4.82) - 7

First, calculate (-4.82)²:

(-4.82)² ≈ 23.2324

Now, substitute this back into the expression:

3(23.2324) + 13(-4.82) - 7 ≈ 69.6972 - 62.66 - 7 ≈ 0.0372

Again, this is very close to zero, indicating that our solution is accurate within the margin of error due to rounding.

Another way to verify the solutions is to use the relationships between the roots and coefficients of a quadratic equation. For a quadratic equation ax² + bx + c = 0, the sum of the roots (x₁ + x₂) is equal to -b/a, and the product of the roots (x₁ * x₂) is equal to c/a. Let's check these relationships with our solutions:

  • Sum of roots: x₁ + x₂ ≈ 0.49 + (-4.82) = -4.33

  • -b/a = -13/3 ≈ -4.33

  • Product of roots: x₁ * x₂ ≈ 0.49 * (-4.82) ≈ -2.36

  • c/a = -7/3 ≈ -2.33

These values are also very close, further confirming the accuracy of our solutions. The slight differences are again due to rounding errors. By using both methods of verification—substitution into the original equation and checking the relationships between roots and coefficients—we can be highly confident that our solutions for the equation 3x² + 13x - 7 = 0 are correct to two decimal places.

Conclusion

In this article, we've thoroughly explored the process of solving the quadratic equation 3x² + 13x - 7 = 0 to two decimal places. We began by understanding the general form of a quadratic equation and identifying the coefficients a, b, and c. We then introduced the quadratic formula, a powerful tool for finding the solutions of any quadratic equation, and discussed the significance of the discriminant in determining the nature of the solutions. By carefully applying the quadratic formula, we found the solutions to be approximately 0.49 and -4.82. We also provided a step-by-step calculation breakdown to ensure clarity and accuracy in the process. Finally, we verified these solutions using two methods: substitution back into the original equation and checking the relationships between the roots and coefficients.

Mastering the solution of quadratic equations is a fundamental skill in algebra, and the quadratic formula is an indispensable tool in this endeavor. By understanding the underlying concepts and practicing the application of the formula, you can confidently tackle a wide range of quadratic equation problems. The ability to solve quadratic equations is not only crucial for success in mathematics but also has applications in various fields, including physics, engineering, and economics. This comprehensive guide has equipped you with the knowledge and skills to solve quadratic equations effectively and accurately.