What Are The Solutions Of X2 8 5x

Solving the Quadratic Equation: x² + 8 = 5x

The seemingly simple equation, x² + 8 = 5x, presents a fantastic opportunity to explore various methods for solving quadratic equations. Understanding these methods not only helps in solving this specific equation but also equips you with valuable tools for tackling more complex quadratic problems. This comprehensive guide will walk you through several approaches, explaining each step in detail and highlighting the strengths and weaknesses of each method.

Understanding Quadratic Equations

Before diving into the solutions, let's establish a foundational understanding. A quadratic equation is an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. Our equation, x² + 8 = 5x, needs to be rearranged into this standard form before we can apply various solution techniques.

Rearranging the Equation

To begin, let's rearrange x² + 8 = 5x into the standard quadratic form:

x² - 5x + 8 = 0

Now that we have the equation in the standard form, we can proceed with different solution methods.

Method 1: Factoring

Factoring is a straightforward method, but it's only applicable if the quadratic expression can be easily factored into two linear expressions. Let's see if we can factor x² - 5x + 8. We're looking for two numbers that add up to -5 (the coefficient of x) and multiply to 8 (the constant term).

Unfortunately, there are no such integer pairs. The factors of 8 are (1, 8), (2, 4), (-1, -8), and (-2, -4). None of these pairs add up to -5. This means that the quadratic expression x² - 5x + 8 is not easily factorable using integers. Therefore, factoring is not a suitable method for solving this particular equation. However, understanding this process is crucial as it's often the quickest method when applicable.

Method 2: Quadratic Formula

The quadratic formula is a powerful tool that works for all quadratic equations, regardless of whether they are easily factorable. The formula is derived from completing the square and provides the solutions directly. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Where 'a', 'b', and 'c' are the coefficients from the standard quadratic equation ax² + bx + c = 0. In our equation, x² - 5x + 8 = 0, we have:

• a = 1
• b = -5
• c = 8

Substituting these values into the quadratic formula:

x = [-(-5) ± √((-5)² - 4 * 1 * 8)] / (2 * 1) x = [5 ± √(25 - 32)] / 2 x = [5 ± √(-7)] / 2

Notice that we have a negative number under the square root. This indicates that the solutions are complex numbers, involving the imaginary unit 'i', where i² = -1.

Therefore, the solutions are:

x = [5 + i√7] / 2 and x = [5 - i√7] / 2

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 easily factored. Let's apply this method to our equation:

x² - 5x + 8 = 0

1. Move the constant term to the right side:

x² - 5x = -8

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

x² - 5x + 25/4 = -8 + 25/4

3. Factor the left side as a perfect square trinomial:

(x - 5/2)² = -8 + 25/4 = -32/4 + 25/4 = -7/4

4. Take the square root of both sides:

x - 5/2 = ±√(-7/4) = ±(i√7)/2

5. Solve for x:

x = 5/2 ± (i√7)/2

This gives us the same complex solutions as the quadratic formula:

x = [5 + i√7] / 2 and x = [5 - i√7] / 2

Understanding the Complex Solutions

The fact that we obtained complex solutions highlights an important aspect of quadratic equations: not all quadratic equations have real number solutions. Complex solutions arise when the discriminant (b² - 4ac) is negative. The discriminant provides crucial information about the nature of the roots (solutions) of a quadratic equation:

• b² - 4ac > 0: Two distinct real roots.
• b² - 4ac = 0: One real root (a repeated root).
• b² - 4ac < 0: Two complex conjugate roots.

In our case, b² - 4ac = (-5)² - 4(1)(8) = 25 - 32 = -7, which is negative. This confirms the presence of two complex conjugate roots.

Graphical Representation

Visualizing the equation graphically can provide further insight. The equation x² - 5x + 8 = 0 represents a parabola. Since the parabola does not intersect the x-axis (where y = 0), there are no real roots. The complex roots represent points in the complex plane, not on the real number line.

Conclusion: A Comprehensive Approach to Solving Quadratic Equations

Solving the seemingly simple equation x² + 8 = 5x has revealed a wealth of information about quadratic equations and their solution methods. We've explored factoring (which proved unsuitable in this instance), the universally applicable quadratic formula, and the method of completing the square. All three methods led to the same pair of complex conjugate solutions. Understanding the discriminant and its implications is key to interpreting the nature of the solutions. Finally, a graphical representation helps visualize why there are no real roots for this specific equation. Mastering these techniques provides a strong foundation for tackling more challenging mathematical problems. Remember that each method offers a unique perspective and understanding the strengths and limitations of each method is crucial for effective problem-solving.