Solving The Quadratic Equation 3t² - T - 1 = 0 Exact Solutions

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In the realm of mathematics, quadratic equations hold a significant place. They appear in various fields, from physics and engineering to economics and computer science. Understanding how to solve them is a fundamental skill for anyone pursuing these disciplines. In this comprehensive guide, we will delve into the process of solving the quadratic equation 3t² - t - 1 = 0 using different methods, ensuring we arrive at exact answers, devoid of decimals. This exploration aims to provide a clear, step-by-step approach, suitable for both beginners and those seeking a refresher.

Understanding Quadratic Equations

Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and x is the variable. The coefficients a, b, and c play crucial roles in determining the nature and value of the solutions, also known as roots, of the equation. These roots represent the values of x that satisfy the equation, making the left-hand side equal to zero.

The Quadratic Formula: A Universal Solver

One of the most reliable methods for solving quadratic equations is the quadratic formula. This formula provides a direct way to find the roots of any quadratic equation, regardless of the complexity of its coefficients. The quadratic formula is derived from the process of completing the square, a technique used to rewrite the quadratic equation in a more manageable form. The formula itself is expressed as:

x = (-b ± √(b² - 4ac)) / 2a

Where:

  • x represents the roots of the equation.
  • a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
  • The symbol ± indicates that there are two possible solutions, one obtained by adding the square root term and the other by subtracting it.
  • The expression b² - 4ac under the square root is known as the discriminant. The discriminant plays a crucial role in determining the nature of the roots.

Applying the Quadratic Formula to 3t² - t - 1 = 0

Now, let's apply the quadratic formula to our specific equation, 3t² - t - 1 = 0. First, we need to identify the coefficients a, b, and c:

  • a = 3
  • b = -1
  • c = -1

Next, we substitute these values into the quadratic formula:

t = (-(-1) ± √((-1)² - 4 * 3 * -1)) / (2 * 3)

Simplifying the expression step-by-step:

  1. t = (1 ± √(1 + 12)) / 6
  2. t = (1 ± √13) / 6

Thus, the two roots of the equation are:

  • t₁ = (1 + √13) / 6
  • t₂ = (1 - √13) / 6

These are the exact solutions to the quadratic equation 3t² - t - 1 = 0, expressed in terms of the square root of 13. Since we are asked for exact answers, we leave the solutions in this form, rather than approximating them as decimals.

Exploring Alternative Methods: Completing the Square

While the quadratic formula provides a direct solution, understanding other methods can offer deeper insights into the nature of quadratic equations. One such method is completing the square. This technique involves manipulating the quadratic equation to create a perfect square trinomial, which can then be easily solved.

Steps for Completing the Square

  1. Divide by the leading coefficient (a): If the coefficient of the t² term (in our case, a = 3) is not 1, divide the entire equation by a. This gives us: t² - (1/3)t - (1/3) = 0

  2. Move the constant term to the right side: Add 1/3 to both sides of the equation: t² - (1/3)t = 1/3

  3. Complete the square: Take half of the coefficient of the t term (which is -1/3), square it, and add it to both sides. Half of -1/3 is -1/6, and squaring it gives us 1/36. So, we add 1/36 to both sides: t² - (1/3)t + 1/36 = 1/3 + 1/36

  4. Factor the left side: The left side is now a perfect square trinomial, which can be factored as: (t - 1/6)² = 1/3 + 1/36

  5. Simplify the right side: Find a common denominator for the fractions on the right side: (t - 1/6)² = 12/36 + 1/36 (t - 1/6)² = 13/36

  6. Take the square root of both sides: Remember to include both the positive and negative square roots: t - 1/6 = ±√(13/36) t - 1/6 = ±√13 / 6

  7. Isolate t: Add 1/6 to both sides: t = 1/6 ± √13 / 6

  8. Combine terms: Express the solutions as single fractions: t = (1 ± √13) / 6

We arrive at the same solutions as with the quadratic formula: t₁ = (1 + √13) / 6 and t₂ = (1 - √13) / 6.

The Discriminant: Nature of the Roots

As mentioned earlier, the discriminant, b² - 4ac, provides valuable information about the nature of the roots of a quadratic equation. By analyzing the discriminant, we can determine whether the equation has two distinct real roots, one repeated real root, or two complex roots.

  • If b² - 4ac > 0: The equation has two distinct real roots. This means there are two different real numbers that satisfy the equation.
  • If b² - 4ac = 0: The equation has one repeated real root. This means there is only one real number that satisfies the equation, and it appears twice as a solution.
  • If b² - 4ac < 0: The equation has two complex roots. This means the solutions involve the imaginary unit i (where i² = -1), and there are no real number solutions.

For our equation, 3t² - t - 1 = 0, the discriminant is:

(-1)² - 4 * 3 * -1 = 1 + 12 = 13

Since 13 > 0, the equation has two distinct real roots, which we have already found using the quadratic formula and completing the square.

Factoring: A Simpler Approach (When Possible)

While the quadratic formula and completing the square are universally applicable, factoring can be a quicker method when the quadratic expression can be easily factored. Factoring involves rewriting the quadratic expression as a product of two binomials. However, not all quadratic equations are easily factorable, especially when the roots are irrational or complex.

In our case, the equation 3t² - t - 1 = 0 does not factor easily using integer coefficients. This is because the roots involve the square root of 13, which is an irrational number. Factoring is most effective when the roots are rational numbers.

Choosing the Right Method

Selecting the most appropriate method for solving a quadratic equation depends on the specific equation and your personal preference. Here's a general guideline:

  • Quadratic Formula: This is the most versatile method and can be used for any quadratic equation. It's particularly useful when the equation is not easily factorable.
  • Completing the Square: This method is valuable for understanding the derivation of the quadratic formula and for transforming quadratic expressions into vertex form. However, it can be more time-consuming than the quadratic formula for solving equations.
  • Factoring: This is the quickest method when the equation can be easily factored. It's often the preferred approach for simpler equations with integer coefficients.

Conclusion

Solving quadratic equations is a fundamental skill in mathematics with applications across various disciplines. In this guide, we have explored different methods for solving the equation 3t² - t - 1 = 0, emphasizing the importance of obtaining exact answers. We applied the quadratic formula, the most reliable method, and also demonstrated the process of completing the square, providing a deeper understanding of the underlying principles. We also discussed the role of the discriminant in determining the nature of the roots and briefly touched upon factoring, a simpler approach for certain equations.

By mastering these techniques, you will be well-equipped to tackle a wide range of quadratic equations and apply them effectively in various mathematical and real-world contexts. Remember to practice regularly and choose the method that best suits the specific equation you are solving. Understanding the strengths and limitations of each method will enhance your problem-solving abilities and deepen your appreciation for the elegance and power of mathematics.

This comprehensive guide aims to provide a clear and accessible explanation of how to solve the quadratic equation 3t² - t - 1 = 0. By understanding the concepts and practicing the techniques, you can confidently solve quadratic equations and apply them in various mathematical and real-world contexts. Remember, mathematics is not just about finding the right answer; it's about understanding the process and developing a logical and analytical approach to problem-solving. Keep exploring, keep practicing, and keep learning!