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In the realm of mathematics, the expression X 2 6X 5 might seem like a simple algebraic equation, but it holds a wealth of applications and insights. This equation, which can be rewritten as X^2 - 6X + 5, is a quadratic equation that can be solved using various methods. Understanding how to solve and interpret this equation is crucial for students and professionals alike, as it forms the foundation for more complex mathematical concepts.

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Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, typically written in the form ax^2 + bx + c = 0. In the case of X 2 6X 5, we have a = 1, b = -6, and c = 5. Quadratic equations are fundamental in algebra and have numerous applications in fields such as physics, engineering, and economics.

Solving the Equation

There are several methods to solve a quadratic equation, including factoring, completing the square, and using the quadratic formula. Let's explore each method in detail.

Factoring

Factoring involves finding two numbers that multiply to give c and add to give b. For the equation X^2 - 6X + 5, we need two numbers that multiply to 5 and add to -6. These numbers are -1 and -5. Therefore, the equation can be factored as:

(X - 1)(X - 5) = 0

Setting each factor equal to zero gives us the solutions:

X - 1 = 0 => X = 1

X - 5 = 0 => X = 5

So, the solutions to the equation X 2 6X 5 are X = 1 and X = 5.

Completing the Square

Completing the square involves manipulating the equation to form a perfect square trinomial. For X^2 - 6X + 5, we first move the constant term to the right side:

X^2 - 6X = -5

Next, we take half of the coefficient of X, square it, and add it to both sides. Half of -6 is -3, and squaring it gives 9:

X^2 - 6X + 9 = -5 + 9

(X - 3)^2 = 4

Taking the square root of both sides gives us:

X - 3 = ±2

Solving for X gives us:

X = 3 + 2 = 5

X = 3 - 2 = 1

Thus, the solutions are X = 1 and X = 5.

Using the Quadratic Formula

The quadratic formula is a general solution for any quadratic equation of the form ax^2 + bx + c = 0. The formula is:

X = [-b ± √(b^2 - 4ac)] / (2a)

For the equation X^2 - 6X + 5, we have a = 1, b = -6, and c = 5. Plugging these values into the formula gives us:

X = [-(-6) ± √((-6)^2 - 4(1)(5))] / (2(1))

X = [6 ± √(36 - 20)] / 2

X = [6 ± √16] / 2

X = [6 ± 4] / 2

This gives us two solutions:

X = (6 + 4) / 2 = 10 / 2 = 5

X = (6 - 4) / 2 = 2 / 2 = 1

Therefore, the solutions are X = 1 and X = 5.

Applications of Quadratic Equations

Quadratic equations have a wide range of applications in various fields. Here are a few examples:

Physics: Quadratic equations are used to describe the motion of objects under constant acceleration, such as projectiles.
Engineering: They are used in designing structures, calculating trajectories, and analyzing electrical circuits.
Economics: Quadratic equations can model profit and loss, supply and demand, and other economic phenomena.
Computer Science: They are used in algorithms for optimization, data fitting, and solving complex problems.

Graphing Quadratic Equations

Graphing a quadratic equation helps visualize its solutions and behavior. The graph of a quadratic equation is a parabola, which is a U-shaped curve. The vertex of the parabola is the point where the curve changes direction. For the equation X^2 - 6X + 5, we can find the vertex by completing the square or using the formula X = -b / (2a).

Using the formula, we have:

X = -(-6) / (2(1)) = 6 / 2 = 3

Substituting X = 3 back into the equation to find the Y value:

Y = (3)^2 - 6(3) + 5 = 9 - 18 + 5 = -4

So, the vertex of the parabola is at the point (3, -4).

Here is a table summarizing the key points of the parabola:

Vertex	X-Intercepts	Y-Intercept
(3, -4)	(1, 0) and (5, 0)	(0, 5)

Graphing the parabola with these points will give a clear visual representation of the equation X 2 6X 5.

📝 Note: The vertex form of a quadratic equation is Y = a(X - h)^2 + k, where (h, k) is the vertex of the parabola. For X^2 - 6X + 5, the vertex form is Y = (X - 3)^2 - 4.

Graphing the parabola with these points will give a clear visual representation of the equation X 2 6X 5.