Introduction
When it comes to solving quadratic equations, one of the most commonly encountered forms is the quadratic trinomial in the form of x^2 – 11x + 4. This type of quadratic equation can be solved using various methods, depending on the specific characteristics of the equation. In this article, we will explore the solutions of the quadratic equation x^2 – 11x + 4 and discuss different approaches to finding these solutions.
Finding the Solutions of x^2 – 11x + 4
To find the solutions of the quadratic equation x^2 – 11x + 4, we can use the quadratic formula, completing the square, or factoring. Each method has its own advantages and is applicable in different scenarios. Let’s delve into each of these methods in detail:
Using the Quadratic Formula
The quadratic formula, x = (-b ± √(b^2 – 4ac)) / 2a, is a general formula that can be used to find the solutions of any quadratic equation in the form of ax^2 + bx + c = 0. In the case of x^2 – 11x + 4, the coefficients a, b, and c are 1, -11, and 4 respectively.
Plugging these values into the quadratic formula, we get:
x = (11 ± √((-11)^2 – 4*1*4)) / 2*1
x = (11 ± √(121 – 16)) / 2
x = (11 ± √105) / 2
Therefore, the solutions of x^2 – 11x + 4 are:
x = (11 + √105) / 2 and x = (11 – √105) / 2
Completing the Square
Completing the square is another method that can be used to find the solutions of a quadratic equation. By completing the square, we can rewrite the quadratic equation in the form (x – h)^2 = k, where h and k are constants.
To complete the square for the quadratic equation x^2 – 11x + 4, we follow these steps:
1. Rewrite the equation as a perfect square trinomial: (x – (11/2))^2 = 25/4
2. Taking the square root of both sides, we get: x – (11/2) = ±√(25/4)
3. Solve for x: x = (11/2) ± 5/2
x = 8 or x = 3
Therefore, the solutions of x^2 – 11x + 4 are x = 8 and x = 3.
Factoring
Factoring is a method that involves breaking down a quadratic equation into its factors to find the values of x that satisfy the equation. For the quadratic equation x^2 – 11x + 4, we can factor it by finding two numbers that multiply to a * c (4) and add up to b (-11).
The factors of 4 are 1, 2, and 4. By testing these factors, we find that -2 and -2 are the numbers that multiply to 4 and add up to -11.
Therefore, the factored form of x^2 – 11x + 4 is:
x^2 – 11x + 4 = (x – 2)(x – 2) = (x – 2)^2
Setting (x – 2)^2 = 0, we find that x = 2 is the only solution.
Conclusion
In conclusion, the solutions of the quadratic equation x^2 – 11x + 4 are x = 8 and x = 3 when using the quadratic formula, x = 8 and x = 3 when completing the square, and x = 2 when factoring. Each method offers a unique approach to solving quadratic equations, and the choice of method depends on the characteristics of the equation at hand.
By understanding the different methods of solving quadratic equations and practicing with various examples, you can enhance your problem-solving skills in mathematics and tackle more complex quadratic equations with confidence. Remember to check your solutions by substituting them back into the original equation to verify their correctness.
