Solve X^2 - 8x - 13 = 0 Find The Value Of T
This article delves into the process of solving quadratic equations, specifically focusing on finding the value of a constant within a solution expressed in a particular form. We will use the example equation x^2 - 8x - 13 = 0 and demonstrate how to determine the value of 't' when one solution is given as 4 + \sqrt{t}. This is a common type of problem encountered in algebra and understanding the methods to solve it is crucial for mastering quadratic equations. We will explore different approaches, including using the quadratic formula and completing the square, to arrive at the solution. By the end of this article, you'll have a solid grasp of how to tackle similar problems and a deeper understanding of quadratic equation solutions.
Understanding Quadratic Equations
Before we dive into the specific problem, let's establish a clear understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. The solutions to a quadratic equation are also known as its roots or zeros. These are the values of 'x' that satisfy the equation. Quadratic equations have at most two real solutions, which can be found using various methods.
There are several methods to solve quadratic equations, each with its own advantages and disadvantages. Some common methods include:
- Factoring: This method involves expressing the quadratic equation as a product of two linear factors. It is a quick and efficient method when the equation can be easily factored. However, not all quadratic equations can be factored easily.
- Completing the Square: This method involves manipulating the equation to form a perfect square trinomial. It is a more general method that can be used to solve any quadratic equation. It also provides a deeper understanding of the structure of quadratic equations.
- Quadratic Formula: This is a formula that directly gives the solutions of a quadratic equation in terms of its coefficients. It is a universal method that can be applied to any quadratic equation, regardless of whether it can be factored or not. The quadratic formula is derived from the method of completing the square.
In the context of the problem at hand, understanding these different methods and their applications will help us choose the most efficient way to find the value of 't'. The given solution form, 4 + \sqrt{t}, hints at the possibility of using the quadratic formula or completing the square, as these methods often yield solutions involving square roots.
Solving x^2 - 8x - 13 = 0
Now, let's tackle the specific quadratic equation: x^2 - 8x - 13 = 0. Our goal is to find the value of 't' given that one solution can be written as 4 + \sqrt{t}. We will use two methods to solve this equation: the quadratic formula and completing the square.
Method 1: Using the Quadratic Formula
The quadratic formula is a powerful tool for solving any quadratic equation in the form ax^2 + bx + c = 0. The formula is given by:
x = (-b ± \sqrt(b^2 - 4ac)) / 2a
In our equation, x^2 - 8x - 13 = 0, we have a = 1, b = -8, and c = -13. Plugging these values into the quadratic formula, we get:
x = (-(-8) ± \sqrt((-8)^2 - 4 * 1 * -13)) / (2 * 1) x = (8 ± \sqrt(64 + 52)) / 2 x = (8 ± \sqrt(116)) / 2
Now, we simplify the square root. Since 116 = 4 * 29, we can write \sqrt(116) as \sqrt(4 * 29) = 2\sqrt(29). Substituting this back into the equation, we get:
x = (8 ± 2\sqrt(29)) / 2
We can simplify this further by dividing both terms in the numerator by 2:
x = 4 ± \sqrt(29)
This gives us two solutions: x = 4 + \sqrt(29) and x = 4 - \sqrt(29). We are given that one solution is in the form 4 + \sqrt{t}. Comparing this with our solution, we can see that t = 29. The quadratic formula allows us to directly calculate the solutions and identify the value of 't' without needing to manipulate the equation further.
Method 2: Completing the Square
Completing the square is another method to solve quadratic equations. It involves rewriting the equation in the form (x - h)^2 = k, where 'h' and 'k' are constants. This form makes it easy to find the solutions by taking the square root of both sides.
To complete the square for the equation x^2 - 8x - 13 = 0, we first focus on the x^2 and x terms. We want to rewrite x^2 - 8x as a perfect square trinomial. To do this, we take half of the coefficient of the x term (-8), which is -4, and square it, which gives us 16. We then add and subtract 16 from the equation:
x^2 - 8x + 16 - 16 - 13 = 0
The first three terms, x^2 - 8x + 16, form a perfect square trinomial, which can be written as (x - 4)^2. So, the equation becomes:
(x - 4)^2 - 16 - 13 = 0 (x - 4)^2 - 29 = 0
Now, we isolate the squared term:
(x - 4)^2 = 29
Taking the square root of both sides, we get:
x - 4 = ±\sqrt(29)
Adding 4 to both sides, we find the solutions:
x = 4 ± \sqrt(29)
Again, we have the two solutions: x = 4 + \sqrt(29) and x = 4 - \sqrt(29). Comparing the solution 4 + \sqrt(29) with the given form 4 + \sqrt{t}, we see that t = 29. Completing the square provides a step-by-step method to transform the equation into a form where the solutions are easily identifiable.
Determining the Value of 't'
Both the quadratic formula and completing the square methods have led us to the same solutions for the equation x^2 - 8x - 13 = 0: x = 4 + \sqrt(29) and x = 4 - \sqrt(29). We were given that one solution can be written as 4 + \sqrt{t}. By comparing this with our solutions, it is clear that:
t = 29
Therefore, the value of the constant 't' is 29. This result confirms that both methods are consistent and provide an accurate solution to the problem. The ability to use multiple methods to solve the same problem is a valuable skill in mathematics, as it allows for verification and a deeper understanding of the underlying concepts.
Key Takeaways and Conclusion
In this article, we tackled the problem of finding the value of 't' in the solution 4 + \sqrt{t} for the quadratic equation x^2 - 8x - 13 = 0. We demonstrated two methods: the quadratic formula and completing the square. Both methods yielded the same solutions, x = 4 + \sqrt(29) and x = 4 - \sqrt(29), leading us to the conclusion that t = 29. This problem highlights several key concepts:
- Quadratic Equations: Understanding the general form of a quadratic equation (ax^2 + bx + c = 0) and the different methods to solve them.
- Quadratic Formula: Applying the quadratic formula (x = (-b ± \sqrt(b^2 - 4ac)) / 2a) to find the solutions of any quadratic equation.
- Completing the Square: Transforming a quadratic equation into the form (x - h)^2 = k to easily find the solutions.
- Solution Forms: Recognizing and comparing different forms of solutions to extract specific values, such as 't' in this case.
By mastering these concepts and methods, you can confidently solve a wide range of quadratic equation problems. The ability to approach a problem using different methods not only helps in finding the solution but also enhances your understanding of the mathematical principles involved.
In conclusion, the value of 't' for the given equation and solution form is 29. This article has provided a comprehensive guide on how to arrive at this answer using the quadratic formula and completing the square, emphasizing the importance of understanding the underlying concepts and methods for solving quadratic equations.
