Finding Real Roots: X^2 - 9x + 18 Using Quadratic Formula

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In this comprehensive guide, we'll explore how to find the real roots of a quadratic equation using the quadratic formula. We'll use the example equation y = x² - 9x + 18 and walk through each step, ensuring you understand the process thoroughly. If you've ever struggled with quadratic equations, this guide is for you. Let's dive in!

Understanding Quadratic Equations

Before we jump into solving, let's quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is:

ax² + bx + c = 0

Where:

  • a, b, and c are constants, with a ≠ 0
  • x represents the variable we want to solve for

In our example, y = x² - 9x + 18, we can rewrite it as x² - 9x + 18 = 0. Comparing this to the general form, we see that a = 1, b = -9, and c = 18. These coefficients are crucial for using the quadratic formula.

The Quadratic Formula: Your Key to Solving

The quadratic formula is a powerful tool that provides the solutions (also known as roots or zeros) for any quadratic equation. It's derived from the method of completing the square and is a reliable way to find the values of x that make the equation true. The formula is given by:

x = (-b ± √(b² - 4ac)) / 2a

This formula might look intimidating at first, but it's quite straightforward once you break it down. The ± symbol means there are two possible solutions: one where you add the square root term and one where you subtract it. These solutions correspond to the points where the parabola represented by the quadratic equation intersects the x-axis.

Applying the Quadratic Formula to Our Example

Now, let's apply the quadratic formula to our equation, x² - 9x + 18 = 0. Remember, we identified a = 1, b = -9, and c = 18. We'll substitute these values into the formula:

  1. Substitute the values: x = (-(-9) ± √((-9)² - 4 * 1 * 18)) / (2 * 1)

  2. Simplify the equation: x = (9 ± √(81 - 72)) / 2 x = (9 ± √9) / 2

  3. Calculate the square root: x = (9 ± 3) / 2

  4. Solve for the two possible values of x: x₁ = (9 + 3) / 2 = 12 / 2 = 6 x₂ = (9 - 3) / 2 = 6 / 2 = 3

Therefore, the real roots of the equation x² - 9x + 18 = 0 are x = 6 and x = 3. These are the x-values where the parabola intersects the x-axis.

Verifying the Roots

It's always a good idea to verify your solutions to ensure they are correct. We can do this by substituting the roots back into the original equation and checking if the equation holds true.

  1. Verify x = 6: (6)² - 9(6) + 18 = 36 - 54 + 18 = 0 (Correct)

  2. Verify x = 3: (3)² - 9(3) + 18 = 9 - 27 + 18 = 0 (Correct)

Since both values satisfy the equation, we can confidently say that our roots are correct.

Interpreting the Roots

The roots of a quadratic equation provide valuable information about the parabola it represents. In the context of the graph y = x² - 9x + 18, the roots x = 3 and x = 6 are the points where the parabola intersects the x-axis. These points are also known as the x-intercepts.

To express these roots as coordinates, we write them as ordered pairs where the y-coordinate is 0. Thus, the roots are (3, 0) and (6, 0). These points are crucial for sketching the graph of the parabola and understanding its behavior.

Choosing the Correct Answer

Based on our calculations, the real roots of the equation y = x² - 9x + 18 are (3, 0) and (6, 0). Looking at the provided options:

A. (0, -3) B. (0, -6) C. (3, 0) D. (6, 0)

The correct answer is a combination of options C and D, as they represent the two real roots we found. However, since we need to select a single option, and both roots are crucial, we should ideally have an option that includes both. If forced to choose, we would select the option that contains at least one of the roots.

Alternative Methods for Finding Roots

While the quadratic formula is a reliable method, there are other ways to find the roots of a quadratic equation. These include:

  • Factoring: This method involves expressing the quadratic equation as a product of two binomials. It's often quicker than the quadratic formula when the equation is easily factorable.
  • Completing the Square: This method involves manipulating the equation to create a perfect square trinomial. It's useful for deriving the quadratic formula and can be used to solve equations.
  • Graphing: By plotting the graph of the quadratic equation, the roots can be visually identified as the points where the parabola intersects the x-axis.

Each method has its advantages and disadvantages, and the best choice depends on the specific equation and your personal preference. However, the quadratic formula is a universal tool that always works, making it a valuable technique to master.

Common Mistakes to Avoid

When working with quadratic equations, it's easy to make mistakes if you're not careful. Here are some common pitfalls to avoid:

  • Incorrectly identifying a, b, and c: Double-check that you have correctly identified the coefficients a, b, and c from the equation.
  • Sign errors: Pay close attention to the signs of the coefficients, especially when substituting them into the quadratic formula.
  • Miscalculating the discriminant (b² - 4ac): The discriminant determines the nature of the roots, so it's crucial to calculate it accurately.
  • Forgetting the ± sign: Remember that the quadratic formula yields two possible solutions, so don't forget to consider both the positive and negative square root.
  • Algebraic errors: Be meticulous with your algebraic manipulations to avoid mistakes in simplifying the equation.

By being aware of these common errors, you can significantly improve your accuracy and confidence in solving quadratic equations.

Real-World Applications of Quadratic Equations

Quadratic equations aren't just abstract mathematical concepts; they have numerous applications in the real world. Here are a few examples:

  • Physics: Quadratic equations are used to describe the motion of projectiles, such as the trajectory of a ball thrown in the air.
  • Engineering: They are used in designing bridges, buildings, and other structures to ensure stability and safety.
  • Economics: Quadratic functions can model cost, revenue, and profit in business scenarios.
  • Computer Graphics: They are used to create curves and surfaces in computer-generated images.
  • Optimization Problems: Quadratic equations can be used to find maximum or minimum values in various optimization problems.

The versatility of quadratic equations makes them a fundamental tool in many fields, highlighting the importance of understanding how to solve them.

Conclusion

Finding the real roots of a quadratic equation using the quadratic formula is a fundamental skill in algebra. By understanding the formula, applying it systematically, and verifying your solutions, you can confidently solve a wide range of quadratic equations. In this guide, we walked through the process step-by-step, using the example equation y = x² - 9x + 18. Remember to pay attention to detail, avoid common mistakes, and practice regularly to master this important technique.

Explore further resources on quadratic equations and mathematical concepts at Khan Academy's Algebra Section.