Solving X² - 11x + 30 = 0 Mastering Quadratic Equations

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This article delves into the process of solving the quadratic equation x² - 11x + 30 = 0. Quadratic equations, which are polynomial equations of the second degree, play a vital role in various fields like mathematics, physics, engineering, and economics. Understanding how to solve them is an essential skill. We'll explore different methods for finding the solutions (also called roots) of this equation, providing a step-by-step explanation to ensure clarity and comprehension. Whether you're a student grappling with algebra or simply someone looking to refresh your math skills, this guide will equip you with the knowledge to confidently tackle quadratic equations.

Understanding Quadratic Equations

Before diving into the solution, it's crucial to grasp the fundamentals of quadratic equations. A quadratic equation is generally expressed in the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable we aim to solve for. The coefficient 'a' cannot be zero, as that would reduce the equation to a linear one. The solutions to a quadratic equation represent the values of 'x' that make the equation true. These solutions can be real numbers, complex numbers, or a combination of both. The number of solutions is determined by the discriminant, which is calculated as b² - 4ac. If the discriminant is positive, there are two distinct real solutions; if it's zero, there is one real solution (a repeated root); and if it's negative, there are two complex solutions.

The equation we're focusing on, x² - 11x + 30 = 0, fits this standard form. Here, a = 1, b = -11, and c = 30. These coefficients will be crucial in applying various methods for solving the equation. One of the most common approaches is factoring, which involves breaking down the quadratic expression into the product of two linear expressions. Another method is using the quadratic formula, a universal tool that provides the solutions regardless of whether the equation can be easily factored. We'll also touch upon completing the square, a technique that transforms the equation into a perfect square trinomial, making it easier to solve. By understanding the underlying principles and applying the appropriate method, solving quadratic equations becomes a manageable and even enjoyable task.

Method 1: Factoring the Quadratic Equation

Factoring is a powerful technique for solving quadratic equations, especially when the equation can be expressed as the product of two binomials. This method relies on finding two numbers that satisfy two conditions: their product equals the constant term (c), and their sum equals the coefficient of the linear term (b). In our equation, x² - 11x + 30 = 0, we need to find two numbers that multiply to 30 and add up to -11. These numbers are -5 and -6, since (-5) * (-6) = 30 and (-5) + (-6) = -11.

Once we've identified these numbers, we can rewrite the middle term (-11x) as the sum of -5x and -6x. This allows us to factor the quadratic expression by grouping. The equation becomes x² - 5x - 6x + 30 = 0. Now, we can factor out the common factors from the first two terms and the last two terms. From the first two terms (x² - 5x), we can factor out an 'x', leaving us with x(x - 5). From the last two terms (-6x + 30), we can factor out a -6, resulting in -6(x - 5). Notice that both terms now have a common factor of (x - 5).

Combining these factored terms, we get (x - 5)(x - 6) = 0. This is the factored form of the quadratic equation. For the product of two factors to be zero, at least one of them must be zero. Therefore, we set each factor equal to zero and solve for 'x'. This gives us two equations: x - 5 = 0 and x - 6 = 0. Solving these simple linear equations, we find the solutions to be x = 5 and x = 6. These are the roots of the quadratic equation x² - 11x + 30 = 0. Factoring is an efficient method when applicable, providing a straightforward path to finding the solutions.

Method 2: Using the Quadratic Formula

The quadratic formula is a universal tool for solving quadratic equations of the form ax² + bx + c = 0. It provides the solutions regardless of whether the equation can be easily factored. The formula is given by:

x = [-b ± √(b² - 4ac)] / 2a

This formula uses the coefficients 'a', 'b', and 'c' from the quadratic equation to directly calculate the values of 'x' that satisfy the equation. The '±' symbol indicates that there are generally two solutions, one obtained by using the plus sign and the other by using the minus sign.

For our equation, x² - 11x + 30 = 0, we have a = 1, b = -11, and c = 30. Substituting these values into the quadratic formula, we get:

x = [-(-11) ± √((-11)² - 4 * 1 * 30)] / (2 * 1)

Simplifying this expression, we first address the terms inside the square root. (-11)² is 121, and 4 * 1 * 30 is 120. So, the expression under the square root becomes 121 - 120 = 1. The square root of 1 is 1. Now, the equation becomes:

x = [11 ± 1] / 2

This gives us two possible solutions. For the plus sign, we have x = (11 + 1) / 2 = 12 / 2 = 6. For the minus sign, we have x = (11 - 1) / 2 = 10 / 2 = 5. Thus, the solutions to the equation x² - 11x + 30 = 0 using the quadratic formula are x = 5 and x = 6, which match the solutions we found by factoring. The quadratic formula is a reliable method that can be applied to any quadratic equation, making it an indispensable tool in solving these types of problems.

Method 3: Completing the Square

Completing the square is another method for solving quadratic equations. It involves transforming the equation into a form where one side is a perfect square trinomial. This method is particularly useful when the equation is not easily factorable or when understanding the structure of quadratic equations is desired. To complete the square, we start with the equation x² - 11x + 30 = 0 and rearrange it by moving the constant term to the right side:

x² - 11x = -30

Next, we need to add a value to both sides of the equation that will make the left side a perfect square trinomial. A perfect square trinomial is an expression that can be factored into the form (x + k)² or (x - k)², where 'k' is a constant. To find this value, we take half of the coefficient of the 'x' term (which is -11), square it, and add the result to both sides of the equation. Half of -11 is -11/2, and squaring it gives us (-11/2)² = 121/4.

Adding 121/4 to both sides of the equation, we get:

x² - 11x + 121/4 = -30 + 121/4

Now, the left side is a perfect square trinomial, which can be factored as (x - 11/2)². To simplify the right side, we need to find a common denominator. -30 can be written as -120/4, so the right side becomes -120/4 + 121/4 = 1/4. The equation now looks like this:

(x - 11/2)² = 1/4

To solve for 'x', we take the square root of both sides:

√(x - 11/2)² = ±√(1/4)

This simplifies to:

x - 11/2 = ±1/2

Now, we have two equations to solve: x - 11/2 = 1/2 and x - 11/2 = -1/2. For the first equation, we add 11/2 to both sides: x = 1/2 + 11/2 = 12/2 = 6. For the second equation, we add 11/2 to both sides: x = -1/2 + 11/2 = 10/2 = 5. Therefore, the solutions to the equation x² - 11x + 30 = 0 using the method of completing the square are x = 5 and x = 6, consistent with the results obtained through factoring and the quadratic formula. Completing the square provides a deeper understanding of the structure of quadratic equations and is a valuable technique in various mathematical contexts.

Verifying the Solutions

After solving a quadratic equation, it's crucial to verify the solutions to ensure accuracy. This process involves substituting the obtained values of 'x' back into the original equation to check if they satisfy the equation. For the equation x² - 11x + 30 = 0, we found two solutions: x = 5 and x = 6.

Let's first verify x = 5. Substituting this value into the equation, we get:

(5)² - 11(5) + 30 = 25 - 55 + 30 = 0

The equation holds true, confirming that x = 5 is indeed a solution.

Next, let's verify x = 6. Substituting this value into the equation, we get:

(6)² - 11(6) + 30 = 36 - 66 + 30 = 0

Again, the equation holds true, confirming that x = 6 is also a solution. This verification step is essential because it helps catch any errors made during the solving process, such as incorrect factoring or misapplication of the quadratic formula. By substituting the solutions back into the original equation, we can confidently confirm that the values we found are the correct roots of the quadratic equation.

Conclusion

In conclusion, we've explored three different methods for solving the quadratic equation x² - 11x + 30 = 0: factoring, using the quadratic formula, and completing the square. Each method provides a unique approach to finding the solutions, and understanding all three enhances your problem-solving toolkit. Factoring is an efficient method when the quadratic expression can be easily factored into two binomials. The quadratic formula is a universal tool that can be applied to any quadratic equation, regardless of its factorability. Completing the square not only provides the solutions but also offers a deeper understanding of the structure of quadratic equations.

For the equation x² - 11x + 30 = 0, all three methods yielded the same solutions: x = 5 and x = 6. We also emphasized the importance of verifying the solutions by substituting them back into the original equation, ensuring accuracy and building confidence in your problem-solving abilities. Mastering the techniques for solving quadratic equations is a fundamental skill in mathematics, with applications spanning various fields. By understanding the different methods and practicing their application, you can confidently tackle a wide range of quadratic equation problems.

  • Solving quadratic equations
  • Quadratic equation solutions
  • Factoring quadratic equations
  • Quadratic formula
  • Completing the square
  • Roots of quadratic equations
  • Algebra
  • Mathematics
  • Equation solving
  • Polynomial equations