by Solving quadratic equations is a fundamental skill in algebra, providing the basis for more complex mathematical and scientific studies. The equation x^2 – 11x + 28 = 0 is an example of a quadratic equation, which is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants. Let’s solve the given quadratic equation step-by-step, exploring different methods to find its roots. The roots of a quadratic equation are the values of x that make the equation true.
The Quadratic Formula Method
The most universal method to solve quadratic equations is using the quadratic formula:
[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} ]
For our equation, x^2 – 11x + 28 = 0, the coefficients are a = 1, b = -11, and c = 28. Plugging these values into the quadratic formula gives:
[ x = \frac{-(-11) \pm \sqrt{(-11)^2 – 4 \cdot 1 \cdot 28}}{2 \cdot 1} ]
[ x = \frac{11 \pm \sqrt{121 – 112}}{2} ]
[ x = \frac{11 \pm \sqrt{9}}{2} ]
[ x = \frac{11 \pm 3}{2} ]
Thus, we find two possible values for x:
[ x_1 = \frac{11 + 3}{2} = 7 ]
[ x_2 = \frac{11 – 3}{2} = 4 ]
So, the solutions to the equation x^2 – 11x + 28 = 0 are x = 7 and x = 4.
Factoring Method
Another approach to solving quadratic equations is factoring, which involves rewriting the quadratic equation as a product of two binomials. This method is particularly efficient when the equation can be easily factored.
For the equation x^2 – 11x + 28 = 0, we look for two numbers that multiply to 28 (the constant term) and add up to -11 (the coefficient of x). These numbers are -7 and -4, since:
[ -7 \cdot -4 = 28 ]
[ -7 + -4 = -11 ]
So, the equation can be factored as:
[ (x – 7)(x – 4) = 0 ]
Setting each factor equal to zero gives us the solutions:
[ x – 7 = 0 \Rightarrow x = 7 ]
[ x – 4 = 0 \Rightarrow x = 4 ]
Therefore, using the factoring method, we again find the solutions to be x = 7 and x = 4.
Completing the Square Method
Completing the square is another method to solve quadratic equations, especially useful when the quadratic formula is not preferred, and the equation is not easily factored. This method involves adding a term to both sides of the equation to complete the square on one side.
For the equation x^2 – 11x + 28 = 0, the process of completing the square involves a few more steps and is less straightforward than the previous methods for this particular equation. Given the simplicity of the quadratic formula and factoring methods for solving this equation, completing the square would be less efficient in this case.
Conclusion
The quadratic equation x^2 – 11x + 28 = 0 has two solutions: x = 7 and x = 4. These solutions can be found using various methods such as the quadratic formula, factoring, or completing the square, each with its own advantages depending on the specific equation. Understanding these methods provides a solid foundation for solving more complex problems in algebra and beyond.
