Solving X In Quadratic Equation X^2 - 12x + 36 = 90
In this article, we will walk through the steps to solve for x in the quadratic equation x² - 12x + 36 = 90. Quadratic equations are a fundamental concept in algebra, and mastering the techniques to solve them is crucial for various mathematical and real-world applications. This equation can be solved using several methods, including factoring, completing the square, and the quadratic formula. We will focus on completing the square and using the square root property to find the solutions. Understanding these methods will empower you to tackle similar problems with confidence.
Quadratic equations are polynomial equations of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠0. The solutions to a quadratic equation, also known as the roots or zeros, are the values of x that satisfy the equation. These roots can be real or complex numbers, and there are several techniques to find them. The given equation, x² - 12x + 36 = 90, is a quadratic equation, and our goal is to find the values of x that make this equation true.
To solve the quadratic equation x² - 12x + 36 = 90, we will employ the method of completing the square. This technique involves manipulating the equation to create a perfect square trinomial on one side, making it easier to isolate x and find its values. Before diving into the steps, it's essential to understand why completing the square is a valuable method. It not only helps in solving quadratic equations but also provides a foundation for understanding other mathematical concepts, such as conic sections and calculus. By mastering this technique, you'll gain a deeper understanding of the structure of quadratic equations and their solutions. Additionally, completing the square is a versatile method that can be used even when factoring is not straightforward, making it a reliable tool in your mathematical arsenal. Let's now proceed with the steps to solve the given equation.
Step 1: Rewrite the Equation
First, we want to rewrite the equation to set it equal to zero. This is a standard form for solving quadratic equations. We achieve this by subtracting 90 from both sides of the equation:
x² - 12x + 36 - 90 = 0
Simplifying, we get:
x² - 12x - 54 = 0
This step is crucial because it sets the stage for applying various solution methods, including factoring, completing the square, and the quadratic formula. By setting the equation equal to zero, we can easily identify the coefficients a, b, and c, which are essential for these methods. In this case, a = 1, b = -12, and c = -54. With the equation in this standard form, we are now prepared to move on to the next step, which involves manipulating the equation to create a perfect square trinomial.
Step 2: Complete the Square
Completing the square involves transforming the quadratic expression into a perfect square trinomial, which can be factored as (x - h)². To do this, we need to add and subtract a value that makes the left side a perfect square. The general formula for completing the square is to add and subtract (b/2)² to the quadratic expression. In our equation, b = -12, so (b/2)² = (-12/2)² = (-6)² = 36. Notice that we already have +36 in our equation x² - 12x - 54 = 0, but it's associated with the constant term. Instead, we'll focus on the x² and -12x terms.
We rewrite the equation as:
(x² - 12x) - 54 = 0
Now, we need to add and subtract (12/2)² = 36 inside the parenthesis to complete the square for the expression inside the parenthesis. However, since we already have -54 as the constant term, we'll move it to the right side of the equation first:
x² - 12x = 54
Now, we add (12/2)² = 36 to both sides:
x² - 12x + 36 = 54 + 36
The left side is now a perfect square trinomial, and the right side simplifies to:
x² - 12x + 36 = 90
This step is the core of the completing the square method. By adding and subtracting the appropriate value, we create an expression that can be easily factored into a squared binomial. This transformation simplifies the equation, making it much easier to solve for x. In the next step, we will factor the perfect square trinomial and proceed to isolate x.
Step 3: Factor the Perfect Square Trinomial
The left side of the equation, x² - 12x + 36, is a perfect square trinomial and can be factored as (x - 6)². So, the equation becomes:
(x - 6)² = 90
Factoring the perfect square trinomial simplifies the equation significantly. This step transforms a complex quadratic expression into a simpler form, which is a squared binomial. This form is much easier to work with when solving for x. The recognition of the perfect square trinomial and its factorization is a crucial skill in algebra, and it's widely used in various mathematical contexts. In this case, by factoring the trinomial, we have successfully reduced the quadratic equation to a form where we can easily apply the square root property to isolate x. This brings us closer to finding the solutions for the equation.
Step 4: Apply the Square Root Property
To solve for x, we take the square root of both sides of the equation:
√(x - 6)² = ±√90
This simplifies to:
x - 6 = ±√90
The square root property is a fundamental concept in algebra that allows us to solve equations where a variable expression is squared. By taking the square root of both sides, we introduce the ± sign, which accounts for both the positive and negative roots of the number. This step is essential because quadratic equations often have two solutions, and considering both the positive and negative square roots ensures that we find all possible values of x that satisfy the equation. In this case, applying the square root property to (x - 6)² = 90 gives us x - 6 = ±√90, which is a crucial step towards isolating x and finding the solutions.
Step 5: Simplify the Square Root
We can simplify √90 by factoring out the perfect square factors. 90 can be factored as 9 × 10, where 9 is a perfect square. So, √90 = √(9 × 10) = √9 × √10 = 3√10.
Substituting this back into our equation, we get:
x - 6 = ±3√10
Simplifying the square root is a crucial step in solving equations involving radicals. By factoring out perfect square factors, we can express the square root in its simplest form, making it easier to work with. In this case, simplifying √90 to 3√10 allows us to express the solutions for x in a more concise and manageable form. This step demonstrates an understanding of radical simplification, which is a valuable skill in algebra and beyond. With the square root simplified, we are now one step closer to finding the final solutions for x.
Step 6: Solve for x
Finally, we isolate x by adding 6 to both sides of the equation:
x = 6 ± 3√10
This gives us two solutions for x:
x = 6 + 3√10 and x = 6 - 3√10
Isolating x is the final step in solving the equation. By adding 6 to both sides, we get the values of x that satisfy the original quadratic equation. The solutions, x = 6 + 3√10 and x = 6 - 3√10, represent the two points where the quadratic function intersects the x-axis. These solutions are real numbers, and they provide a complete answer to the problem. Understanding how to isolate the variable and obtain the solutions is a fundamental skill in algebra, and it's essential for solving various mathematical problems.
Therefore, the solutions for x in the equation x² - 12x + 36 = 90 are x = 6 ± 3√10. This corresponds to option A. By completing the square and using the square root property, we have successfully found the values of x that satisfy the given quadratic equation. This method provides a systematic approach to solving quadratic equations, ensuring accurate results. Mastering these techniques is essential for success in algebra and related fields.
The final answer is (A) x = 6 ± 3√10.
