Solving Quadratic Equation X² - 12x + 36 = 90 A Step By Step Guide
Introduction
In the realm of mathematics, solving equations is a fundamental skill. Among various types of equations, quadratic equations hold a significant place due to their wide applications in various fields, including physics, engineering, and economics. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The equation presented, x² - 12x + 36 = 90, is a classic example of a quadratic equation, and in this comprehensive guide, we will explore the step-by-step process of solving it. This exploration will not only enhance your understanding of quadratic equations but also equip you with the skills to tackle similar problems with confidence. The key to mastering quadratic equations lies in understanding the underlying principles and applying the appropriate techniques. Whether you're a student preparing for an exam or simply someone looking to brush up on your math skills, this guide will provide you with the knowledge and tools necessary to conquer quadratic equations. Let's embark on this mathematical journey together and unlock the secrets of solving for x.
Understanding the Quadratic Equation
Before diving into the solution, it's crucial to understand the nature of the quadratic equation we're dealing with: x² - 12x + 36 = 90. A quadratic equation is characterized by its highest power of the variable being 2. This equation can be solved using various methods, including factoring, completing the square, and using the quadratic formula. Each method has its own advantages and is suitable for different types of quadratic equations. Factoring is often the quickest method when the equation can be easily factored, while completing the square is a more general method that can be used for any quadratic equation. The quadratic formula is a powerful tool that provides a direct solution for any quadratic equation, regardless of its complexity. In this particular case, we will explore the method of completing the square and also touch upon how the quadratic formula could be applied. Understanding the structure of a quadratic equation is the first step towards solving it. Recognizing the coefficients and the constant term helps in choosing the most appropriate method for solving the equation. Moreover, understanding the discriminant, which is a part of the quadratic formula, can provide insights into the nature of the roots of the equation, whether they are real or complex, distinct or repeated.
Step 1: Rearrange the Equation
The first step in solving any quadratic equation is to bring it into the standard form, which is ax² + bx + c = 0. In our case, the equation is x² - 12x + 36 = 90. To get it into the standard form, we need to subtract 90 from both sides of the equation. This process maintains the equality while rearranging the terms to fit the standard quadratic form. Subtracting 90 from both sides, we get:
x² - 12x + 36 - 90 = 90 - 90
This simplifies to:
x² - 12x - 54 = 0
Now, the equation is in the standard quadratic form, where a = 1, b = -12, and c = -54. Having the equation in this form is crucial for applying various solution methods, such as completing the square or using the quadratic formula. The coefficients a, b, and c play a significant role in determining the nature of the roots and the steps involved in solving the equation. Rearranging the equation is a fundamental step that sets the stage for the subsequent steps in the solution process. It ensures that we are working with a standard form that allows us to apply the appropriate techniques effectively.
Step 2: Completing the Square
Completing the square is a powerful technique for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial on one side. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. In our equation, x² - 12x - 54 = 0, we will complete the square for the expression x² - 12x. To do this, we take half of the coefficient of the x term (which is -12), square it, and add it to both sides of the equation. Half of -12 is -6, and (-6)² is 36. Notice that the first three terms on the left side of the equation, x² - 12x + 36, already form a perfect square trinomial, which can be factored as (x - 6)². This is a crucial observation that simplifies the process of completing the square. However, to illustrate the general method, we can still add 36 to both sides of the equation:
x² - 12x - 54 + 36 = 0 + 36
This simplifies to:
x² - 12x + 36 = 90
Now, we can rewrite the left side as a squared binomial:
(x - 6)² = 90
Completing the square is a versatile method that can be used to solve any quadratic equation, even those that are not easily factorable. It provides a systematic approach to finding the roots of the equation and is particularly useful when dealing with equations that have irrational or complex roots.
Step 3: Taking the Square Root
Now that we have the equation in the form (x - 6)² = 90, we can proceed by taking the square root of both sides. This step is crucial for isolating x and ultimately finding its value. When taking the square root, it's essential to remember that there are two possible solutions: a positive square root and a negative square root. This is because both the positive and negative values, when squared, will result in the same positive number. Taking the square root of both sides, we get:
√(x - 6)² = ±√90
This simplifies to:
x - 6 = ±√90
The ± sign indicates that we have two possible solutions to consider. This is a characteristic feature of quadratic equations, which can have up to two distinct roots. The square root of 90 can be further simplified by factoring out the largest perfect square factor, which is 9. We can rewrite √90 as √(9 * 10), which simplifies to 3√10. This simplification makes the solution more concise and easier to work with. Taking the square root is a fundamental step in solving quadratic equations and other types of equations involving squares or higher powers. It allows us to undo the squaring operation and move closer to isolating the variable we are trying to solve for.
Step 4: Simplify the Square Root
In the previous step, we arrived at the equation x - 6 = ±√90. To simplify the solution further, we need to simplify the square root term, √90. As mentioned earlier, we can factor 90 as 9 * 10, where 9 is a perfect square. This allows us to rewrite the square root as:
√90 = √(9 * 10) = √9 * √10 = 3√10
This simplification is crucial for expressing the solution in its most concise and understandable form. Simplifying square roots involves identifying and factoring out perfect square factors from the radicand (the number under the square root sign). This process makes the solution easier to interpret and compare with other solutions. In many mathematical contexts, simplified solutions are preferred as they provide a clearer representation of the answer. The ability to simplify square roots is a valuable skill in algebra and calculus, as it often arises in various problem-solving scenarios. Understanding the properties of square roots and how to manipulate them is essential for mastering mathematical concepts.
Step 5: Isolate x
Now that we have simplified the square root, our equation looks like this:
x - 6 = ±3√10
To isolate x, we need to add 6 to both sides of the equation. This step will give us the value of x in terms of the simplified square root. Adding 6 to both sides, we get:
x - 6 + 6 = 6 ± 3√10
This simplifies to:
x = 6 ± 3√10
This equation represents two distinct solutions for x: one where we add 3√10 to 6, and another where we subtract 3√10 from 6. Isolating the variable is a fundamental step in solving any equation, regardless of its type. It involves performing inverse operations to get the variable by itself on one side of the equation. In this case, we used addition to undo the subtraction and isolate x. The final expression for x provides the solutions to the original quadratic equation. These solutions are often referred to as the roots of the equation.
Step 6: Final Solutions
Therefore, the solutions for x in the equation x² - 12x + 36 = 90 are:
x = 6 + 3√10 and x = 6 - 3√10
These are the two values of x that satisfy the original equation. We can write this in a more concise form as:
x = 6 ± 3√10
This notation represents both solutions simultaneously. The solutions are irrational numbers, as they involve the square root of 10, which cannot be expressed as a simple fraction. Quadratic equations can have two distinct real solutions, one repeated real solution, or two complex solutions. The nature of the solutions depends on the discriminant, which is the part of the quadratic formula under the square root sign (b² - 4ac). In this case, the discriminant is positive, indicating that we have two distinct real solutions. The solutions we obtained are the values of x where the parabola represented by the quadratic equation intersects the x-axis. These solutions are crucial in various applications, such as finding the points of intersection between curves, determining the maximum or minimum values of a quadratic function, and solving real-world problems that can be modeled using quadratic equations.
Alternative Method: Using the Quadratic Formula
While we solved the equation by completing the square, it's worth noting that the quadratic formula provides a direct method for finding the solutions of any quadratic equation in the form ax² + bx + c = 0. The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
In our equation, x² - 12x - 54 = 0, we have a = 1, b = -12, and c = -54. Plugging these values into the quadratic formula, we get:
x = (-(-12) ± √((-12)² - 4 * 1 * -54)) / (2 * 1)
x = (12 ± √(144 + 216)) / 2
x = (12 ± √360) / 2
Now, we simplify √360. We can factor 360 as 36 * 10, where 36 is a perfect square:
√360 = √(36 * 10) = √36 * √10 = 6√10
Substituting this back into the equation, we get:
x = (12 ± 6√10) / 2
Dividing both terms in the numerator by 2, we obtain:
x = 6 ± 3√10
As you can see, the quadratic formula yields the same solutions as completing the square. The quadratic formula is a powerful tool that can be used to solve any quadratic equation, regardless of its complexity. It is particularly useful when the equation is not easily factorable or when completing the square is cumbersome.
Conclusion
In this guide, we have demonstrated how to solve the quadratic equation x² - 12x + 36 = 90 using the method of completing the square and also verified the solution using the quadratic formula. We found that the solutions are x = 6 + 3√10 and x = 6 - 3√10, which can be written concisely as x = 6 ± 3√10. Understanding how to solve quadratic equations is a fundamental skill in mathematics, with applications in various fields. Whether you choose to complete the square, use the quadratic formula, or factor the equation, the key is to follow a systematic approach and pay attention to the details. By mastering these techniques, you will be well-equipped to tackle a wide range of mathematical problems involving quadratic equations. Remember, practice is essential for solidifying your understanding and building confidence in your problem-solving abilities. So, continue to explore different types of quadratic equations and apply the methods you've learned to become a proficient problem solver.
The correct answer is A. x = 6 ± 3√10.
