The answer of x2-11x+28=0 is x = 7 and x = 4.
A quadratic equation is a second-degree polynomial equation, which means the highest power of the variable is squared.
The general form of a quadratic equation is: ax^2 + bx + c = 0
where a, b, and c are constants, and ‘x’ represents the variable we’re trying to find.
Factoring Quadratic Equations
One of the most common methods to solve quadratic equations is factoring. Factoring involves breaking down the equation into two binomial expressions that, when multiplied together, equal the original equation.
For the equation, x^2 – 11x + 28 = 0, we can factor it into (x – 7)(x – 4) = 0, which allows us to find the solutions for ‘x.’
(x – 7)(x – 4) = 0
Setting each factor to zero, we get:
x – 7 = 0, and x – 4 = 0
Solving for ‘x’ in both equations, we find two solutions: x = 7 and x = 4.
Using the Quadratic Formula
Another powerful process for solving quadratic equations is the quadratic formula:
x = (-b ± √(b^2 – 4ac)) / 2a
After putting the value of a, b, c we can gate the value x=7 and x=4 from the above equation.
Real Roots and Discriminant
In the context of quadratic equations, the discriminant, which is the value inside the square root of the quadratic formula, helps us determine the nature of the roots. If the discriminant is positive, there are two distinct real roots. If it’s zero, there’s one real root (a repeated root). If it’s negative, there are no real roots (complex roots).
Graphical Representation of Quadratic Equations
Quadratic equations can also be represented graphically as parabolas. The solutions to the equation are the x-values where the graph intersects the x-axis. In the case of x^2 – 11x + 28 = 0, we see that the parabola intersects the x-axis at x = 7 and x = 4, confirming our solutions.
Applications of Quadratic Equations
Quadratic equations are not just theoretical concepts. They have practical applications in various fields, including physics, engineering, economics, and more. They are often used to model real-world phenomena where relationships are inherently quadratic.