Quadratic equations are a fundamental topic in algebra, and they appear in various mathematical and real-world contexts. The equation x2 – 11x + 28 = 0 is a quadratic equation that we’ll explore in this article. We’ll take a step-by-step approach to solve this equation and find its roots.
Understanding the Quadratic Equation
A quadratic equation is a polynomial equation of the second degree, typically written in the form ax² + bx + c = 0, where a, b, and c are coefficients, and x is the variable we’re solving for. In our equation, x² – 11x + 28 = 0, we have:
- a = 1
- b = -11
- c = 28
The primary goal in solving a quadratic equation is to determine the values of x that make the equation true.
Factoring the Quadratic Equation
One method to solve quadratic equations is by factoring. To factor the equation x2 – 11x + 28 = 0, we aim to break it down into two binomials that multiply to equal the original equation. We look for two numbers that multiply to the constant term (28) and add up to the coefficient of the middle term (-11). These numbers are -7 and -4 because (-7) * (-4) = 28, and (-7) + (-4) = -11.
Now, we can rewrite the equation with these numbers:
(x – 7)(x – 4) = 0
Solving for x
We have factored the equation into two parts:
- x – 7 = 0
- x – 4 = 0
Now, we can solve each equation separately to find the values of x:
Solving Equation 1: x – 7 = 0
Add 7 to both sides to isolate x:
x = 7
Solving Equation 2: x – 4 = 0
Add 4 to both sides to isolate x:
x = 4
The Roots of the Quadratic Equation
Now that we’ve solved both equations derived from the factored form, we have two possible values for x:
- x = 7
- x = 4
These values represent the roots or solutions to the quadratic equation x2 – 11x + 28 = 0. In other words, if we substitute these values back into the original equation, it will hold true:
For x = 7:
7² – 11(7) + 28 = 49 – 77 + 28 = 0
For x = 4:
4² – 11(4) + 28 = 16 – 44 + 28 = 0
Both values of x satisfy the equation, making them the roots of the quadratic equation.
In conclusion, we have successfully solved the quadratic equation x2 – 11x + 28 = 0 and found its roots. The solutions are:
- x = 7
- x = 4
These values of x make the equation true and represent the points at which the graph of the quadratic equation intersects the x-axis. Solving quadratic equations is a fundamental skill in algebra, and it has applications in various fields of mathematics, science, engineering, and beyond. Understanding how to approach and solve such equations is essential for problem-solving in these disciplines.