What Are The Solutions To X2 8x 7 0

Solving the Quadratic Equation: x² + 8x + 7 = 0

The equation x² + 8x + 7 = 0 is a classic example of a quadratic equation. Quadratic equations are fundamental in algebra and have wide-ranging applications in various fields, from physics and engineering to economics and computer science. Understanding how to solve these equations is crucial for anyone studying mathematics or related subjects. This comprehensive guide will explore multiple methods for solving x² + 8x + 7 = 0, explaining each step in detail and providing insights into the underlying mathematical principles.

Understanding Quadratic Equations

Before diving into the solutions, let's briefly review the general form of a quadratic equation:

ax² + bx + c = 0

Where:

• a, b, and c are constants (numbers).
• x is the variable we're trying to solve for.
• a ≠ 0 (if a were 0, the equation would be linear, not quadratic).

In our specific case, x² + 8x + 7 = 0, we have:

• a = 1
• b = 8
• c = 7

Method 1: Factoring

Factoring is often the quickest and easiest method for solving quadratic equations, especially when the factors are relatively straightforward. The goal is to rewrite the equation as a product of two binomials.

1. Find two numbers that add up to 'b' (8) and multiply to 'c' (7). In this case, those numbers are 1 and 7 (1 + 7 = 8 and 1 * 7 = 7).

2. Rewrite the equation using these numbers:

   (x + 1)(x + 7) = 0

3. Apply the Zero Product Property: This property states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore:

   x + 1 = 0 or x + 7 = 0

4. Solve for x:

   x = -1 or x = -7

Therefore, the solutions to the equation x² + 8x + 7 = 0 are x = -1 and x = -7.

Method 2: Quadratic Formula

The quadratic formula is a powerful tool that can be used to solve any quadratic equation, regardless of whether it's easily factorable or not. The formula is derived from completing the square and is given by:

x = [-b ± √(b² - 4ac)] / 2a

Let's apply this to our equation:

1. Substitute the values of a, b, and c:

   x = [-8 ± √(8² - 4 * 1 * 7)] / (2 * 1)

2. Simplify the expression:

   x = [-8 ± √(64 - 28)] / 2 x = [-8 ± √36] / 2 x = [-8 ± 6] / 2

3. Solve for the two possible values of x:

   x = (-8 + 6) / 2 = -2 / 2 = -1 x = (-8 - 6) / 2 = -14 / 2 = -7

Again, we arrive at the solutions x = -1 and x = -7. The quadratic formula provides a reliable method even when factoring isn't immediately apparent.

Method 3: Completing the Square

Completing the square is a technique used to manipulate the quadratic equation into a perfect square trinomial, which can then be easily solved.

1. Move the constant term to the right side of the equation:

   x² + 8x = -7

2. Take half of the coefficient of the x term (8/2 = 4), square it (4² = 16), and add it to both sides of the equation:

   x² + 8x + 16 = -7 + 16 x² + 8x + 16 = 9

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

   (x + 4)² = 9

4. Take the square root of both sides:

   x + 4 = ±√9 x + 4 = ±3

5. Solve for x:

   x = -4 + 3 = -1 x = -4 - 3 = -7

Once more, the solutions are x = -1 and x = -7. Completing the square is a valuable technique that provides a deeper understanding of the underlying structure of quadratic equations.

Graphical Representation and Interpretation

The solutions to the quadratic equation x² + 8x + 7 = 0 represent the x-intercepts (points where the graph crosses the x-axis) of the parabola represented by the function y = x² + 8x + 7. Graphing the quadratic function provides a visual representation of the solutions. The parabola will intersect the x-axis at x = -1 and x = -7. This visual confirmation reinforces the algebraic solutions found using the previously discussed methods. The vertex of the parabola, representing the minimum point of the function, can also be found using the formula x = -b/2a, which in this case is x = -4. Substituting this back into the equation gives the y-coordinate of the vertex, y = -9.

Applications of Quadratic Equations

Quadratic equations are not merely abstract mathematical concepts; they have numerous real-world applications. Here are a few examples:

• Physics: Calculating projectile motion, analyzing the trajectory of objects under the influence of gravity.
• Engineering: Designing bridges, buildings, and other structures; analyzing stress and strain on materials.
• Economics: Modeling supply and demand curves, optimizing production levels.
• Computer Graphics: Creating curved lines and shapes, representing 3D objects.

Advanced Concepts and Extensions

For those seeking a deeper understanding, here are some related advanced topics:

• Complex Roots: Some quadratic equations have solutions that involve imaginary numbers (involving the square root of -1). These are called complex roots.
• Discriminant: The discriminant (b² - 4ac) in the quadratic formula determines the nature of the roots. If it's positive, there are two distinct real roots; if it's zero, there's one real root (a repeated root); and if it's negative, there are two complex roots.
• Higher-Order Polynomials: Quadratic equations are a special case of polynomials. Solving higher-order polynomials (cubic, quartic, etc.) involves more advanced techniques.

Conclusion

Solving the quadratic equation x² + 8x + 7 = 0, using factoring, the quadratic formula, or completing the square, consistently yields the solutions x = -1 and x = -7. Understanding these methods and their underlying principles is essential for success in algebra and numerous related fields. The graphical representation provides a visual confirmation of these solutions and highlights the practical applications of quadratic equations in various disciplines. Exploring advanced concepts like the discriminant and complex roots further solidifies one’s understanding of the rich mathematical landscape surrounding quadratic equations.