For Which Value Of Is

Solving for x: A Deep Dive into the Equation x² - 5x + 6 = 0

This article explores the solution to the quadratic equation x² - 5x + 6 = 0, explaining multiple methods for finding the values of 'x' that satisfy the equation. Because of that, we'll break down the underlying mathematical principles, provide step-by-step solutions, and address frequently asked questions. That's why understanding this seemingly simple equation provides a strong foundation for tackling more complex algebraic problems. This guide is designed for students of all levels, from those just beginning their algebra journey to those looking for a refresher.

Understanding Quadratic Equations

Before jumping into the solution, let's clarify what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in this case, 'x') is 2. And 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. Our equation, x² - 5x + 6 = 0, fits this form with a = 1, b = -5, and c = 6.

The solutions to a quadratic equation are also known as its roots or zeros. But these are the values of 'x' that make the equation true. A quadratic equation can have up to two real roots, one real root (a repeated root), or two complex roots (involving imaginary numbers).

Method 1: Factoring

Factoring is a powerful and often the quickest method for solving quadratic equations, particularly when the factors are easily identifiable. The goal is to rewrite the quadratic expression as a product of two linear expressions.

Here's how to factor x² - 5x + 6 = 0:

1. Find two numbers that add up to 'b' (-5) and multiply to 'c' (6). These numbers are -2 and -3. (-2) + (-3) = -5 and (-2) * (-3) = 6.

2. Rewrite the equation using these numbers: (x - 2)(x - 3) = 0

3. Set each factor equal to zero and solve for 'x':

   • x - 2 = 0 => x = 2
   • x - 3 = 0 => x = 3

Which means, the solutions to the equation x² - 5x + 6 = 0 are x = 2 and x = 3.

Method 2: Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation, regardless of whether it can be easily factored. The formula is derived from completing the square method and provides the roots directly.

The quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

Let's apply it to our equation:

1. Identify the values of a, b, and c: a = 1, b = -5, c = 6

2. Substitute these values into the quadratic formula:

x = [-(-5) ± √((-5)² - 4 * 1 * 6)] / (2 * 1)

3. Simplify the expression:

x = [5 ± √(25 - 24)] / 2

x = [5 ± √1] / 2

x = [5 ± 1] / 2

4. Solve for the two possible values of x:

• x = (5 + 1) / 2 = 3
• x = (5 - 1) / 2 = 2

Again, the solutions are x = 2 and x = 3.

Method 3: Completing the Square

Completing the square is a method used to manipulate the quadratic equation into a perfect square trinomial, allowing for easy solution. While not always the fastest, it's an important technique to understand as it forms the basis for deriving the quadratic formula.

Here's how to solve x² - 5x + 6 = 0 by completing the square:

For more on this topic, read our article on which statement is true about opening issued boxes of ammunition or check out which two stars have the most similar temperatures and luminosity.

1. Move the constant term to the right side of the equation: x² - 5x = -6

2. Take half of the coefficient of 'x' (-5), square it ((-5/2)² = 25/4), and add it to both sides of the equation:

x² - 5x + 25/4 = -6 + 25/4

3. Rewrite the left side as a perfect square:

(x - 5/2)² = 1/4

4. Take the square root of both sides:

x - 5/2 = ±√(1/4) = ±1/2

5. Solve for 'x':

• x = (5/2) + (1/2) = 3
• x = (5/2) - (1/2) = 2

Once again, we arrive at the solutions x = 2 and x = 3.

Graphical Representation

The solutions to the quadratic equation represent the x-intercepts (points where the graph crosses the x-axis) of the parabola defined by the function y = x² - 5x + 6. If you were to graph this function, you would see the parabola intersecting the x-axis at x = 2 and x = 3. This visual representation confirms our algebraic solutions.

The Discriminant and Nature of Roots

The expression inside the square root in the quadratic formula (b² - 4ac) is called the discriminant. The discriminant helps determine the nature of the roots:

• If b² - 4ac > 0: The equation has two distinct real roots.
• If b² - 4ac = 0: The equation has one real root (a repeated root).
• If b² - 4ac < 0: The equation has two complex roots (involving imaginary numbers).

In our case, b² - 4ac = (-5)² - 4 * 1 * 6 = 1 > 0, indicating two distinct real roots, which we've already found to be 2 and 3.

Frequently Asked Questions (FAQ)

Q: Can I use any method to solve a quadratic equation?

A: While the quadratic formula works for all quadratic equations, factoring is often faster and simpler when the equation factors easily. Completing the square is a valuable technique for understanding the derivation of the quadratic formula and is sometimes useful in other mathematical contexts.

Q: What if the quadratic equation doesn't have real solutions?

A: If the discriminant (b² - 4ac) is negative, the solutions will be complex numbers involving the imaginary unit 'i' (where i² = -1).

Q: Why are there two solutions?

A: A quadratic equation represents a parabola, which can intersect the x-axis at two points, each corresponding to a solution.

Q: What if 'a' is zero?

A: If 'a' is zero, the equation is no longer quadratic; it becomes a linear equation, and solving techniques for linear equations should be applied.

Q: How can I check my solutions?

A: Substitute the calculated values of 'x' back into the original equation. Now, if both sides of the equation are equal, the solutions are correct. Take this: for x = 2: (2)² - 5(2) + 6 = 4 - 10 + 6 = 0. The same holds true for x = 3.

Conclusion

Solving the quadratic equation x² - 5x + 6 = 0 has demonstrated three effective methods: factoring, the quadratic formula, and completing the square. But each approach provides valuable insight into the underlying mathematical principles. On the flip side, understanding these methods not only solves this specific equation but equips you with the skills to tackle a wide range of quadratic equations and opens the door to more advanced algebraic concepts. Remember to choose the method that best suits the equation and your comfort level, and always check your solutions! The beauty of mathematics lies in its versatility and the multiple paths available to reach the correct answer.