Solving X² - 12x + 36 = 0 Using The Quadratic Formula A Step-by-Step Guide

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Understanding Quadratic Equations

At its core, a quadratic equation is a polynomial equation of the second degree. This means it includes at least one term that is squared. The standard form of a quadratic equation is expressed as:

ax² + bx + c = 0

where 'a', 'b', and 'c' are coefficients, and 'x' represents the variable we aim to solve for. The coefficient 'a' cannot be zero; otherwise, the equation would become linear rather than quadratic. Understanding this standard form is crucial because it sets the stage for applying various solution methods, including the quadratic formula, which we will explore in detail.

  • 'a': This coefficient is the numerical value associated with the x² term. It dictates the parabola's direction (whether it opens upwards or downwards) and its width. A larger absolute value of 'a' results in a narrower parabola, while a smaller value leads to a wider one. The sign of 'a' determines the parabola's concavity—positive for upward-opening and negative for downward-opening.
  • 'b': The 'b' coefficient is linked to the x term and influences the parabola's axis of symmetry. This axis is a vertical line that divides the parabola into two mirror-image halves. The 'b' coefficient, in conjunction with 'a', determines the horizontal position of this axis. It plays a pivotal role in locating the vertex of the parabola, which is the point where the parabola changes direction.
  • 'c': This constant term represents the y-intercept of the parabola. It's the point where the parabola intersects the y-axis. The 'c' value provides a direct indication of the parabola's vertical shift—a higher 'c' value moves the parabola upwards, and a lower value shifts it downwards. It also affects the position of the vertex and the overall appearance of the graph.

The solutions to a quadratic equation, also known as roots or zeros, are the x-values that make the equation true. Graphically, these solutions represent the points where the parabola intersects the x-axis. A quadratic equation can have two distinct real solutions, one repeated real solution, or two complex solutions. The nature of these solutions is determined by the discriminant, a key component of the quadratic formula, which we'll explore later. The quadratic formula allows us to find these solutions regardless of the complexity of the equation, making it a versatile tool in algebra.

Introduction to the Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions (or roots) of any quadratic equation, regardless of its complexity. It is particularly useful when factoring is difficult or impossible. The formula is derived from the process of completing the square on the general form of a quadratic equation and is expressed as follows:

x = (-b ± √(b² - 4ac)) / 2a

Where:

  • 'x' represents the solutions (roots) of the quadratic equation.
  • 'a', 'b', and 'c' are the coefficients from the standard form of the quadratic equation: ax² + bx + c = 0.
  • The '±' symbol indicates that there are two possible solutions: one using addition and one using subtraction.
  • The expression inside the square root, b² - 4ac, is known as the discriminant. It plays a crucial role in determining the nature of the solutions.

The discriminant (b² - 4ac) provides valuable information about the nature of the roots of the quadratic equation:

  • If b² - 4ac > 0: The equation has two distinct real solutions. This means the parabola intersects the x-axis at two different points.
  • If b² - 4ac = 0: The equation has exactly one real solution (a repeated root). This means the parabola touches the x-axis at only one point, which is the vertex of the parabola.
  • If b² - 4ac < 0: The equation has two complex solutions. This means the parabola does not intersect the x-axis.

The discriminant is a key indicator of the type of solutions you can expect, guiding you in your approach to solving the quadratic equation. Understanding the quadratic formula and its components, including the discriminant, empowers you to tackle a wide range of quadratic equations effectively. The formula is not just a mathematical tool but a gateway to understanding the behavior and solutions of quadratic functions, making it an essential concept in algebra and beyond.

Step-by-Step Solution Using the Quadratic Formula for x² - 12x + 36 = 0

To effectively apply the quadratic formula, it's essential to follow a systematic approach. Let's break down the process step by step using the example equation x² - 12x + 36 = 0. This method ensures accuracy and clarity in your solution.

  1. Identify the Coefficients: The first step is to correctly identify the coefficients 'a', 'b', and 'c' from the standard form of the quadratic equation, ax² + bx + c = 0. In our example, x² - 12x + 36 = 0, we have:

    • a = 1 (the coefficient of x²)
    • b = -12 (the coefficient of x)
    • c = 36 (the constant term) Accurate identification of these coefficients is crucial as they will be directly substituted into the quadratic formula. Mistakes in this step will lead to incorrect solutions, so double-check your values before proceeding. Correctly recognizing the signs of the coefficients, especially 'b', is also vital for the subsequent calculations.

  2. Write Down the Quadratic Formula: To avoid errors and ensure proper application, write down the quadratic formula explicitly:

    x = (-b ± √(b² - 4ac)) / 2a

    Having the formula in front of you as you work through the problem helps prevent mistakes and reinforces your understanding of the process. It also serves as a reference point, ensuring that you substitute the values correctly in the subsequent steps. This practice is particularly useful when dealing with complex quadratic equations or when you're learning the formula.

  3. Substitute the Coefficients: Now, substitute the identified values of 'a', 'b', and 'c' into the quadratic formula. For our equation, x² - 12x + 36 = 0, the substitution looks like this:

    x = (-(-12) ± √((-12)² - 4 * 1 * 36)) / (2 * 1)

    Pay close attention to signs during this step, especially when substituting negative values. Use parentheses to ensure correct calculations. This substitution step is a critical juncture where precision is paramount to avoid errors in the final solution. Double-check each value as you substitute it into the formula to maintain accuracy.

  4. Simplify the Expression: After substituting the coefficients, the next step is to simplify the expression within the quadratic formula. This involves performing the arithmetic operations in the correct order. Start by simplifying the expression inside the square root, known as the discriminant:

    • Calculate (-12)² which equals 144.
    • Calculate 4 * 1 * 36 which equals 144.
    • Subtract the second result from the first: 144 - 144 = 0. Now, simplify the rest of the equation:

    x = (12 ± √0) / 2

    x = (12 ± 0) / 2

    This simplification is a critical step in arriving at the solution and requires careful attention to detail. Simplifying the discriminant first can provide insights into the nature of the solutions (real, repeated, or complex).

  5. Solve for x: The final step is to solve for 'x' by performing the remaining calculations. In our simplified equation, x = (12 ± 0) / 2, we have:

    • x = (12 + 0) / 2 = 12 / 2 = 6
    • x = (12 - 0) / 2 = 12 / 2 = 6

    In this case, we obtain a single repeated root: x = 6. This means the quadratic equation has only one solution. When the discriminant is zero, as in this case, the quadratic equation has exactly one real solution, indicating that the parabola touches the x-axis at only one point. This final step is where all the previous steps culminate in the solution, highlighting the importance of accuracy and precision throughout the process.

By following these steps methodically, you can confidently solve quadratic equations using the quadratic formula. Each step builds upon the previous one, ensuring a clear and accurate path to the solution. This systematic approach not only helps in finding the correct answer but also enhances your understanding of the quadratic formula and its applications.

Detailed Solution for x² - 12x + 36 = 0

Now, let's walk through the detailed solution of the given quadratic equation, x² - 12x + 36 = 0, using the quadratic formula. This will provide a clear understanding of how the formula is applied step-by-step, reinforcing the process we outlined earlier.

  1. Identify the Coefficients: As we discussed, the first step involves identifying the coefficients 'a', 'b', and 'c' from the equation. In the given equation, x² - 12x + 36 = 0, we have:

    • a = 1
    • b = -12
    • c = 36 These coefficients are the foundation for applying the quadratic formula, and ensuring their accuracy is paramount for a correct solution.

  2. Write Down the Quadratic Formula: The next step is to write down the quadratic formula itself. This helps in visualizing the structure and ensures that we substitute the values in the correct places. The formula is:

    x = (-b ± √(b² - 4ac)) / 2a

    Having the formula written down acts as a constant reference, minimizing the chances of error in the subsequent substitution step.

  3. Substitute the Coefficients: Now, we substitute the values of 'a', 'b', and 'c' into the quadratic formula. Substituting a = 1, b = -12, and c = 36, we get:

    x = (-(-12) ± √((-12)² - 4 * 1 * 36)) / (2 * 1)

    This is a crucial step, and attention to detail is essential. Make sure you replace each variable with its correct value, paying particular attention to the signs.

  4. Simplify the Expression: After substitution, the next step is to simplify the expression. This involves simplifying the discriminant (the expression under the square root) and then the entire formula. Let's break it down:

    • Simplify inside the square root:
      • (-12)² = 144
      • 4 * 1 * 36 = 144
      • 144 - 144 = 0 So, the expression under the square root simplifies to 0. Now, the formula looks like this:

    x = (12 ± √0) / 2

    Since √0 = 0, we can further simplify:

    x = (12 ± 0) / 2

    This simplification is crucial for arriving at the solution. A zero discriminant indicates that the quadratic equation has exactly one real solution.

  5. Solve for x: Finally, we solve for 'x'. Since we have ± 0, both the addition and subtraction will yield the same result:

    • x = (12 + 0) / 2 = 12 / 2 = 6
    • x = (12 - 0) / 2 = 12 / 2 = 6

    Thus, the quadratic equation x² - 12x + 36 = 0 has one repeated root: x = 6.

In summary, by carefully identifying the coefficients, writing down the formula, substituting the values, simplifying the expression, and solving for 'x', we successfully found the solution to the given quadratic equation using the quadratic formula. This step-by-step approach not only provides the solution but also reinforces a clear understanding of the method, making it easier to apply in future problems. The fact that we obtained a single repeated root highlights the significance of the discriminant in determining the nature of the solutions.

Alternative Methods and When to Use Them

While the quadratic formula is a reliable method for solving any quadratic equation, it's not always the most efficient. Other methods, such as factoring and completing the square, can be quicker and simpler in certain situations. Understanding these alternative methods and knowing when to apply them can significantly improve your problem-solving efficiency. Additionally, using the graphical method can provide a visual understanding of the solutions.

1. Factoring

Factoring involves expressing the quadratic equation as a product of two binomials. This method is particularly effective when the quadratic equation can be easily factored. For example, consider the equation x² - 5x + 6 = 0. This can be factored as (x - 2)(x - 3) = 0. The solutions are then found by setting each factor equal to zero:

  • x - 2 = 0 => x = 2
  • x - 3 = 0 => x = 3

Factoring is often faster than the quadratic formula when the equation has integer roots and is easily factorable. However, it's not always straightforward, especially when the coefficients are large or the roots are not integers. In such cases, the quadratic formula becomes a more reliable option. To determine if factoring is a viable approach, look for two numbers that multiply to the constant term (c) and add up to the coefficient of the linear term (b). If you can quickly find such numbers, factoring is likely the faster method.

2. Completing the Square

Completing the square is another method for solving quadratic equations. It involves transforming the equation into a perfect square trinomial. This method is particularly useful for deriving the quadratic formula itself and can be efficient for equations where the coefficient of x² (a) is 1 and the coefficient of x (b) is an even number. For example, let's solve x² + 6x + 5 = 0 by completing the square:

  1. Move the constant term to the other side: x² + 6x = -5
  2. Add (b/2)² to both sides: (6/2)² = 9, so x² + 6x + 9 = -5 + 9
  3. Factor the left side as a perfect square: (x + 3)² = 4
  4. Take the square root of both sides: x + 3 = ±2
  5. Solve for x: x = -3 ± 2, which gives x = -1 and x = -5

Completing the square can be more involved than factoring for simple equations but is valuable for understanding the structure of quadratic equations and for situations where you need to rewrite the equation in vertex form. This method is also essential for tackling problems where you need to find the maximum or minimum value of a quadratic function.

3. Graphical Method

The graphical method involves plotting the quadratic equation on a coordinate plane and finding the points where the parabola intersects the x-axis. These points represent the solutions to the equation. This method provides a visual understanding of the solutions and is particularly useful for estimating the roots when they are not integers or when you have access to graphing tools. For example, if you plot the equation y = x² - 4x + 3, the parabola intersects the x-axis at x = 1 and x = 3, which are the solutions to the equation x² - 4x + 3 = 0.

The graphical method is not always precise, especially when the roots are irrational numbers, but it provides a valuable visual check for solutions obtained through other methods. It also helps in understanding the relationship between the roots and the shape of the parabola. The vertex of the parabola, which can be easily identified on the graph, provides additional information about the quadratic function, such as its minimum or maximum value.

When to Use Each Method

  • Use Factoring when the equation is easily factorable with integer roots.
  • Use Completing the Square when the coefficient of x² is 1 and the coefficient of x is even, or when you need to rewrite the equation in vertex form.
  • Use the Quadratic Formula when factoring is difficult or impossible, or when you need a reliable method that works for all quadratic equations.
  • Use the Graphical Method for visual estimation of solutions and to check solutions obtained through other methods.

By mastering these alternative methods and understanding when to use them, you can approach quadratic equations with greater flexibility and efficiency. The quadratic formula remains a powerful tool, but knowing when to use other methods can save time and effort, enhancing your problem-solving skills.

Real-World Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications across various fields. Understanding these applications can help you appreciate the practical significance of quadratic equations and how they are used to model and solve problems in the real world.

1. Physics

In physics, quadratic equations are fundamental in describing the motion of objects under constant acceleration. Projectile motion, which includes the trajectory of a ball thrown in the air or a rocket launched into space, can be modeled using quadratic equations. The equation that describes the height of a projectile over time is a quadratic equation, accounting for factors such as initial velocity, launch angle, and gravitational acceleration.

For instance, the equation h(t) = -16t² + v₀t + h₀ describes the height (h) of an object at time (t), where v₀ is the initial velocity and h₀ is the initial height. The -16t² term represents the effect of gravity. Solving this quadratic equation allows physicists to determine the time it takes for the object to reach a certain height, its maximum height, or its range. The roots of the quadratic equation represent the times when the object is at ground level (h = 0), providing crucial information about the projectile's path.

2. Engineering

Engineers use quadratic equations extensively in designing structures, circuits, and systems. In structural engineering, quadratic equations are used to calculate the load-bearing capacity of beams and arches. The parabolic shape of suspension bridges and arches is described by quadratic equations, which help engineers optimize the structure for stability and strength. The solutions to these equations determine critical parameters such as the maximum load the structure can withstand before failing.

In electrical engineering, quadratic equations are used to analyze circuits and determine the power dissipated in a resistor. The power equation, P = I²R (where P is power, I is current, and R is resistance), involves a squared term, making it a quadratic equation. Solving this equation allows engineers to design circuits that operate efficiently and safely, ensuring that components do not overheat or fail. Quadratic equations are also used in control systems engineering to model and control the behavior of dynamic systems, such as robots and automated machinery.

3. Finance

Quadratic equations also play a significant role in finance, particularly in modeling financial growth and investment returns. Compound interest calculations often involve quadratic equations, especially when dealing with annuities and loans. The future value of an investment or the loan repayment schedule can be determined using quadratic equations that account for the principal amount, interest rate, and time period.

For example, the equation for compound interest, A = P(1 + r/n)^(nt), can be transformed into a quadratic equation when solving for variables such as the interest rate or time period. Financial analysts use these equations to evaluate investment opportunities, assess risk, and plan for future financial needs. Quadratic equations are also used in option pricing models and other advanced financial calculations, making them an essential tool for finance professionals.

4. Computer Graphics

In computer graphics, quadratic equations are used to create curves and surfaces. Bezier curves and splines, which are fundamental elements in computer-aided design (CAD) and animation, are often defined using polynomial equations, including quadratic equations. These curves are used to create smooth shapes and realistic models in 3D graphics applications. The solutions to quadratic equations help determine the shape and position of these curves, ensuring that the final image or animation appears as intended.

Quadratic equations are also used in ray tracing algorithms, which simulate the way light interacts with objects in a scene. Determining whether a ray of light intersects with a surface often involves solving a quadratic equation. The solutions indicate the points of intersection, allowing the algorithm to calculate the color and brightness of each pixel in the final image. This is crucial for creating realistic and visually appealing graphics in video games, movies, and other applications.

5. Other Applications

Beyond these core areas, quadratic equations appear in various other fields:

  • Sports: Modeling the trajectory of a ball in sports like baseball or golf.
  • Architecture: Designing arches and parabolic structures.
  • Environmental Science: Modeling population growth and decay.

In conclusion, quadratic equations are a powerful tool with wide-ranging applications in science, engineering, finance, and technology. Their ability to model curved paths, optimize structures, and solve for unknown quantities makes them indispensable in many real-world scenarios. By understanding the applications of quadratic equations, you can better appreciate their relevance and importance in solving practical problems.

Conclusion

In conclusion, the quadratic formula is a fundamental tool in mathematics for solving quadratic equations. It provides a reliable method for finding the solutions, or roots, of any quadratic equation, regardless of its complexity. While alternative methods like factoring and completing the square can be more efficient in certain situations, the quadratic formula is a versatile and universally applicable technique.

Throughout this article, we've explored the significance of quadratic equations, the derivation and application of the quadratic formula, and the importance of understanding the discriminant. We've also examined alternative methods for solving quadratic equations and highlighted numerous real-world applications where these equations play a crucial role. By mastering the quadratic formula and related concepts, you gain a powerful tool for tackling a wide range of mathematical problems.

The step-by-step approach to solving quadratic equations using the quadratic formula involves:

  1. Identifying the coefficients 'a', 'b', and 'c' from the equation.
  2. Writing down the quadratic formula to ensure correct application.
  3. Substituting the coefficients into the formula with careful attention to signs.
  4. Simplifying the expression, including the discriminant, to determine the nature of the roots.
  5. Solving for 'x' by performing the necessary arithmetic operations.

This systematic approach not only helps in finding the correct solutions but also reinforces your understanding of the underlying concepts. Understanding the discriminant (b² - 4ac) is particularly important, as it provides insights into whether the quadratic equation has two distinct real solutions, one repeated real solution, or two complex solutions.

Beyond the mechanics of solving equations, we've also seen how quadratic equations are used in various fields such as physics, engineering, finance, and computer graphics. From modeling projectile motion to designing structures and analyzing financial growth, quadratic equations provide a mathematical framework for understanding and solving real-world problems. This practical relevance underscores the importance of mastering quadratic equations and the quadratic formula.

In summary, the quadratic formula is an essential tool in mathematics and a gateway to understanding a wide range of real-world phenomena. By understanding its derivation, application, and significance, you can enhance your problem-solving skills and appreciate the beauty and power of quadratic equations.