When dealing with quadratic equations, finding the solutions is crucial in mathematics and science. In this article, we will discuss the solutions of the equation X^2 – 8 = 5X. We will explore different methods to solve this quadratic equation and understand the concepts behind them. Let’s dive in!
Understanding the Quadratic Equation X^2 – 8 = 5X
Before we delve into the solutions of the quadratic equation X^2 – 8 = 5X, let’s break down the equation and understand its components:
- Quadratic Equation: A quadratic equation is a second-degree polynomial equation that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants, and x represents the variable.
- X^2 Term: In this equation, the term X^2 represents the quadratic term, which is the highest degree term in the equation.
- Constant Term: The constant term in this equation is -8, which is independent of the variable X.
- Linear Term: The linear term in this equation is 5X, which consists of a coefficient (5) and the variable X.
Solving the Quadratic Equation X^2 – 8 = 5X
There are several methods to solve a quadratic equation, including factoring, completing the square, and using the quadratic formula. Let’s explore each method for finding the solutions of X^2 – 8 = 5X:
Method 1: Factoring
Factoring is a common method used to solve quadratic equations by breaking down the equation into simpler factors. To solve X^2 – 8 = 5X by factoring, we need to rewrite the equation in the form ax^2 + bx + c = 0 and factor the equation. Here’s how:
X^2 – 8 = 5X
X^2 – 5X – 8 = 0
(X – 8)(X + 1) = 0
By setting each factor to zero, we find the solutions:
- X – 8 = 0
- X = 8
- X + 1 = 0
- X = -1
Therefore, the solutions of X^2 – 8 = 5X are X = 8 and X = -1.
Method 2: Completing the Square
Completing the square is another method used to solve quadratic equations by creating a perfect square trinomial. To solve X^2 – 8 = 5X by completing the square, we need to rewrite the equation in the form ax^2 + bx + c = 0 and complete the square. Here’s how:
X^2 – 8 = 5X
X^2 – 5X – 8 = 0
X^2 – 5X = 8
X^2 – 5X + (5/2)^2 = 8 + (5/2)^2
(X – 5/2)^2 = 49/4
Taking the square root of both sides, we get:
X – 5/2 = ±7/2
X = 5/2 ± 7/2
X = 6 or X = -1
Therefore, the solutions of X^2 – 8 = 5X are X = 6 and X = -1.
Method 3: Quadratic Formula
The quadratic formula is a formula used to find the solutions of a quadratic equation of the form ax^2 + bx + c = 0. The formula is given as:
X = (-b ± sqrt(b^2 – 4ac)) / 2a
For the equation X^2 – 8 = 5X, we have a = 1, b = -5, and c = -8. Plugging these values into the quadratic formula, we get:
X = (5 ± sqrt(25 + 32)) / 2
X = (5 ± sqrt(57)) / 2
Therefore, the solutions of X^2 – 8 = 5X are:
- X = (5 + sqrt(57)) / 2
- X = (5 – sqrt(57)) / 2
Conclusion
In conclusion, the solutions of the quadratic equation X^2 – 8 = 5X are X = 8, X = -1, X = 6, and X = -1. By utilizing different methods such as factoring, completing the square, and the quadratic formula, we can efficiently find the roots of the equation. Understanding these methods is essential for solving quadratic equations and mastering the principles of algebra.
