Solving X² - 10x + 25 = 35 A Step By Step Guide
In the realm of algebra, solving equations is a fundamental skill. Quadratic equations, with their squared variable, often present a unique challenge. This article provides a comprehensive, step-by-step solution to the equation x² - 10x + 25 = 35, exploring various methods and concepts involved. By understanding the underlying principles, you'll be well-equipped to tackle similar problems with confidence.
Understanding Quadratic Equations
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 roots or zeros, are the values of x that satisfy the equation. These solutions represent the points where the parabola defined by the quadratic equation intersects the x-axis.
There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. The choice of method often depends on the specific form of the equation and personal preference. In this article, we will primarily focus on completing the square and using the square root property, as these methods are particularly well-suited for the given equation.
Method 1: Completing the Square
Completing the square is a technique used to rewrite a quadratic expression in the form (x + p)² + q, where p and q are constants. This form allows us to easily isolate x and solve for its value. Let's apply this method to our equation, x² - 10x + 25 = 35.
Step 1: Recognize the Perfect Square Trinomial
The left side of the equation, x² - 10x + 25, is a perfect square trinomial. This means it can be factored into the form (x - a)². In this case, we have:
x² - 10x + 25 = (x - 5)²
This is because (x - 5)² = (x - 5)(x - 5) = x² - 5x - 5x + 25 = x² - 10x + 25. Recognizing perfect square trinomials allows for a more efficient solving process, simplifying the equation significantly.
Step 2: Rewrite the Equation
Now we can rewrite the original equation as:
(x - 5)² = 35
This transformation is crucial because it sets the stage for isolating x by taking the square root of both sides. This step elegantly simplifies the problem, making it much easier to manage.
Step 3: Apply the Square Root Property
To solve for x, we take the square root of both sides of the equation. Remember that when taking the square root, we must consider both the positive and negative roots:
√(x - 5)² = ±√35
This yields:
x - 5 = ±√35
The square root property is a fundamental tool in solving quadratic equations. It acknowledges that there are two possible solutions, one positive and one negative, when dealing with squares. This step is vital for finding all possible values of x.
Step 4: Isolate x
Finally, we isolate x by adding 5 to both sides of the equation:
x = 5 ± √35
This gives us two possible solutions for x: x = 5 + √35 and x = 5 - √35. Isolating x is the ultimate goal in solving any equation, and this step clearly presents the solutions to the quadratic equation.
Method 2: Direct Application of the Square Root Property
Alternatively, we can directly apply the square root property after recognizing the perfect square trinomial. This method is a streamlined version of completing the square, especially when the quadratic expression is already in a suitable form.
Step 1: Recognize and Rewrite
As before, recognize that x² - 10x + 25 is (x - 5)². Rewrite the equation as:
(x - 5)² = 35
This step is identical to the first step in Method 1, highlighting the importance of recognizing perfect square trinomials. This recognition allows for a more direct path to the solution.
Step 2: Apply the Square Root Property
Take the square root of both sides, remembering both positive and negative roots:
√(x - 5)² = ±√35
This results in:
x - 5 = ±√35
Again, the square root property is crucial here, ensuring that all possible solutions are considered. This step simplifies the equation to a point where isolating x becomes straightforward.
Step 3: Isolate x
Add 5 to both sides to isolate x:
x = 5 ± √35
This yields the same solutions as Method 1: x = 5 + √35 and x = 5 - √35. The consistency of the solution across different methods reinforces its validity.
Analyzing the Solutions
Both methods have led us to the same solutions: x = 5 + √35 and x = 5 - √35. These are the exact values of x that satisfy the original equation. Approximating these values using a calculator, we find:
x ≈ 5 + 5.92 ≈ 10.92
x ≈ 5 - 5.92 ≈ -0.92
These approximations provide a numerical sense of the solutions, allowing for a practical understanding of their magnitude. However, it's important to retain the exact solutions (5 + √35 and 5 - √35) for precision in mathematical contexts.
Connecting to the Options
Looking at the given options:
A. x = 10 ± √35 B. x = 5 ± √35 C. x = 5 ± 2√5 D. x = 10 ± 2√5
We can clearly see that option B. x = 5 ± √35 matches our solutions. This confirms our calculations and demonstrates the effectiveness of the chosen methods.
Key Concepts and Takeaways
- Perfect Square Trinomials: Recognizing and factoring perfect square trinomials simplifies quadratic equations.
- Completing the Square: This method transforms a quadratic expression into a form that facilitates solving for x.
- Square Root Property: Remember to consider both positive and negative roots when taking the square root of both sides of an equation.
- Exact vs. Approximate Solutions: While approximations are useful, maintaining exact solutions (with radicals) is crucial for mathematical accuracy.
- Multiple Solution Methods: There are often multiple ways to solve a quadratic equation. Choosing the most efficient method can save time and effort.
Practice Problems
To solidify your understanding, try solving these similar quadratic equations:
- x² + 6x + 9 = 16
- x² - 8x + 16 = 25
- x² + 4x + 4 = 9
By practicing these problems, you'll gain confidence and proficiency in solving quadratic equations using completing the square and the square root property. These skills are fundamental in algebra and will serve you well in more advanced mathematical studies.
Conclusion
Solving the equation x² - 10x + 25 = 35 demonstrates the power of algebraic techniques like completing the square and applying the square root property. By recognizing perfect square trinomials and understanding the principles behind these methods, we can efficiently find the solutions. This step-by-step guide provides a solid foundation for tackling a wide range of quadratic equations. Remember to practice regularly and explore different approaches to enhance your problem-solving skills in mathematics. The correct answer to the equation x² - 10x + 25 = 35 is indeed x = 5 ± √35, which corresponds to option B.
