X 2 10x 25 0

Decoding the Mystery: x² + 10x + 25 = 0 and the World of Quadratic Equations

This article delves into the seemingly simple equation x² + 10x + 25 = 0, exploring its solution, the underlying mathematical concepts, and its broader significance within the realm of quadratic equations. Understanding this seemingly basic equation unlocks a deeper understanding of more complex algebraic problems and their applications in various fields. We will cover several methods of solving this equation and explain the mathematical principles behind them, making this a valuable resource for students and anyone interested in strengthening their algebra skills.

Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero (otherwise, it wouldn't be a quadratic equation). Our focus equation, x² + 10x + 25 = 0, is a specific example of this general form, with a = 1, b = 10, and c = 25.

Quadratic equations are fundamental in mathematics and have wide-ranging applications in physics, engineering, economics, and computer science. They are used to model various phenomena, from projectile motion to the growth of populations. Mastering the techniques for solving quadratic equations is therefore crucial for success in many scientific and technical fields.

Solving x² + 10x + 25 = 0: Method 1 – Factoring

Factoring is a powerful technique for solving quadratic equations, especially when the equation can be easily factored. This method involves rewriting the equation as a product of two simpler expressions. Let's apply this method to our equation:

x² + 10x + 25 = 0

Observe that the expression x² + 10x + 25 is a perfect square trinomial. This means it can be factored into the square of a binomial. Specifically:

x² + 10x + 25 = (x + 5)(x + 5) = (x + 5)²

Therefore, our equation becomes:

(x + 5)² = 0

To solve for x, we take the square root of both sides:

√(x + 5)² = √0

x + 5 = 0

Subtracting 5 from both sides gives us:

x = -5

Therefore, the solution to the equation x² + 10x + 25 = 0 is x = -5. This is a repeated root, meaning the parabola represented by the equation touches the x-axis at only one point, x = -5.

Solving x² + 10x + 25 = 0: Method 2 – Quadratic Formula

The quadratic formula is a more general method that can be used to solve any quadratic equation, regardless of whether it can be easily factored. The formula is derived from completing the square and provides a direct way to find the solutions. The quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

For our equation, a = 1, b = 10, and c = 25. Substituting these values into the quadratic formula, we get:

x = [-10 ± √(10² - 4 * 1 * 25)] / (2 * 1)

x = [-10 ± √(100 - 100)] / 2

x = [-10 ± √0] / 2

x = -10 / 2

x = -5

Again, we find that the solution to the equation is x = -5. This confirms the result we obtained using the factoring method.

Solving x² + 10x + 25 = 0: Method 3 – Completing the Square

Completing the square is another algebraic technique used to solve quadratic equations. This method involves manipulating the equation to create a perfect square trinomial, which can then be factored easily.

Let's apply this method to our equation:

x² + 10x + 25 = 0

To complete the square, we take half of the coefficient of the x term (10/2 = 5), square it (5² = 25), and add and subtract this value to the left side of the equation:

x² + 10x + 25 - 25 + 25 = 0

Notice that adding and subtracting 25 doesn't change the equation's value. Now, we can rewrite the equation as:

(x² + 10x + 25) = 0

This is a perfect square trinomial, which can be factored as:

(x + 5)² = 0

Taking the square root of both sides and solving for x, as done in the factoring method, gives us x = -5.

The Discriminant and the Nature of Roots

The expression inside the square root in the quadratic formula, b² - 4ac, is called the discriminant. The discriminant determines the nature of the roots (solutions) of the quadratic equation:

• b² - 4ac > 0: The equation has two distinct real roots.
• b² - 4ac = 0: The equation has one repeated real root (as in our case).
• b² - 4ac < 0: The equation has two complex conjugate roots (roots involving the imaginary unit 'i').

In our equation, x² + 10x + 25 = 0, the discriminant is 10² - 4 * 1 * 25 = 0. This indicates that the equation has one repeated real root, which we found to be x = -5.

Graphical Representation

The equation x² + 10x + 25 = 0 represents a parabola. The parabola opens upwards (since the coefficient of x² is positive) and its vertex lies on the x-axis at x = -5. The fact that the parabola touches the x-axis at only one point visually confirms that there is only one repeated root.

Applications of Quadratic Equations

Quadratic equations have numerous applications in various fields:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity.
• Engineering: Designing structures, analyzing stresses and strains in materials.
• Economics: Modeling supply and demand, optimizing production.
• Computer Science: Developing algorithms, solving optimization problems.

Frequently Asked Questions (FAQ)

Q: What does it mean to have a repeated root?

A: A repeated root means that the quadratic equation has only one solution, and that solution appears twice. Graphically, it means the parabola touches the x-axis at only one point.

Q: Can all quadratic equations be solved by factoring?

A: No, not all quadratic equations can be easily factored. The quadratic formula is a more general method that can be used to solve any quadratic equation.

Q: What is the significance of the discriminant?

A: The discriminant tells us the nature of the roots of the quadratic equation – whether they are real or complex, and whether they are distinct or repeated.

Q: What if the coefficient of x² (a) is zero?

A: If 'a' is zero, the equation is no longer quadratic; it becomes a linear equation. A linear equation has only one root, and it can be solved easily by algebraic manipulation.

Conclusion

The equation x² + 10x + 25 = 0, while seemingly simple, provides a valuable foundation for understanding quadratic equations. Through factoring, the quadratic formula, and completing the square, we demonstrated three different methods to solve this equation and arrive at the repeated root x = -5. Understanding the concept of the discriminant helps predict the nature of the roots in any quadratic equation. This knowledge extends far beyond this specific equation, forming a crucial building block for tackling more complex mathematical problems and their real-world applications across diverse fields. The exploration of this single equation opens a gateway to a deeper understanding of the power and versatility of algebra.