Solving the Quadratic Equation x2 2x - 8 0: Methods and Techniques
In this article, we will explore the methods for finding the roots of the quadratic equation x2 2x - 8 0. We will use the quadratic formula and factorization techniques, as well as discuss the underlying methods and processes.
1. Using the Quadratic Formula
The standard quadratic equation can be written as:
ax2 bx c 0
For the given equation, we can identify the coefficients as:
a 1, b 2, and c -8.
The quadratic formula for finding the roots of this equation is:
x [-b plusmn; √(b2 - 4ac)] / 2a
Substituting the values of the coefficients:
x [-2 plusmn; √(22 - 4(1)(-8)))] / 2(1)
The discriminant is calculated as:
b2 - 4ac 4 32 36
Applying the quadratic formula, we get:
x [-2 plusmn; √36] / 2
Solving for the two possible values of x:
x1 (-2 6) / 2 2
x2 (-2 - 6) / 2 -4
Thus, the roots of the equation x2 2x - 8 0 are x 2 and x -4.
2. Factorization Method
The given equation can be written as:
x2 2x - 8 0
This can be factored as:
(x - 4)(x 2) 0
Setting each factor to zero, we get:
x - 4 0 or x 2 0
Solving these equations:
x1 4 and x2 -2
After careful observation and verification, we find the roots to be:
x1 2 and x2 -4
3. Alternative Factorization Methods
There are several ways to factorize the given equation. Here are three such methods:
Method 1:
x2 2x - 8 0
This can be written as:
x2 - 4x 6x - 8 0
Grouping terms, we get:
x(x - 4) 2(x - 4) 0
Factoring out the common term:
(x 2)(x - 4) 0
Thus, the roots are:
x1 -4 and x2 2
Method 2:
x2 2x - 8 0
This can be written as:
x2 - 4x 2x - 8 0
Grouping terms, we get:
x(x - 4) 2(x - 4) 0
Factoring out the common term:
(x 2)(x - 4) 0
Thus, the roots are:
x1 -4 and x2 2
Method 3:
Let the roots be A and B.
Sum of the roots (A B) -2
Product of the roots (A * B) -8
Using the quadratic formula:
A - B plusmn; 6
When A - B 6 and A * B -8:
2A 4, A 2, B -4
When A - B -6 and A * B -8:
2A -8, A -4, B 2
Thus, the roots are:
x1 -4 and x2 2
Conclusion
In this article, we have explored the methods used to solve the quadratic equation x2 2x - 8 0. We have utilized the quadratic formula, factorization techniques, and alternative factorization methods to find the roots of the equation. Each method has its unique advantages and can be applied depending on the specific circumstances. Whether you are a student or a professional, understanding these techniques is essential for solving quadratic equations efficiently and accurately.