Factor X 2 2x 48

Solving the Equation: x² + 2x - 48 = 0

This article will guide you through the process of solving the quadratic equation x² + 2x - 48 = 0. We'll explore multiple methods, including factoring, the quadratic formula, and completing the square. Understanding how to solve quadratic equations is fundamental in algebra and has numerous applications in various fields like physics, engineering, and economics. This comprehensive guide will not only show you how to solve this specific equation but also equip you with the skills to tackle similar problems.

Understanding Quadratic Equations

Before diving into the solution, let's understand what a quadratic equation is. 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 ≠ 0 (if a were 0, it wouldn't be a quadratic equation). In our case, x² + 2x - 48 = 0, we have a = 1, b = 2, and c = -48.

Method 1: Factoring

Factoring is often the quickest and easiest method to solve a quadratic equation, provided the equation can be factored easily. The goal is to find two binomials whose product equals the quadratic expression. We're looking for two numbers that add up to 'b' (2 in our case) and multiply to 'c' (-48).

Let's think about the factors of -48:

• 1 and -48
• 2 and -24
• 3 and -16
• 4 and -12
• 6 and -8
• 8 and -6
• 12 and -4
• 16 and -3
• 24 and -2
• 48 and -1

The pair that adds up to 2 is 8 and -6. Therefore, we can factor the equation as follows:

(x + 8)(x - 6) = 0

This equation is true if either (x + 8) = 0 or (x - 6) = 0. Solving for x in each case gives us:

• x + 8 = 0 => x = -8
• x - 6 = 0 => x = 6

Therefore, the solutions to the equation x² + 2x - 48 = 0 are x = -8 and x = 6.

Method 2: Quadratic Formula

The quadratic formula is a more general method that works for all quadratic equations, even those that are difficult or impossible to factor easily. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Substituting the values from our equation (a = 1, b = 2, c = -48), we get:

x = [-2 ± √(2² - 4 * 1 * -48)] / (2 * 1) x = [-2 ± √(4 + 192)] / 2 x = [-2 ± √196] / 2 x = [-2 ± 14] / 2

This gives us two solutions:

• x = (-2 + 14) / 2 = 12 / 2 = 6
• x = (-2 - 14) / 2 = -16 / 2 = -8

Again, we find the solutions x = 6 and x = -8. This confirms the results obtained through factoring.

Method 3: Completing the Square

Completing the square is another method for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily factored.

1. Move the constant term to the right side: x² + 2x = 48

2. Take half of the coefficient of x (which is 2), square it (1), and add it to both sides: x² + 2x + 1 = 48 + 1 x² + 2x + 1 = 49

3. Factor the left side as a perfect square trinomial: (x + 1)² = 49

4. Take the square root of both sides: x + 1 = ±√49 x + 1 = ±7

5. Solve for x: x = -1 + 7 = 6 x = -1 - 7 = -8

Once again, we arrive at the solutions x = 6 and x = -8.

Graphical Representation

The solutions to the quadratic equation x² + 2x - 48 = 0 represent the x-intercepts (where the graph crosses the x-axis) of the parabola represented by the function y = x² + 2x - 48. If you were to graph this function, you would see that the parabola intersects the x-axis at x = 6 and x = -8. This provides a visual confirmation of our solutions.

Why Multiple Methods?

We explored three different methods to solve the same quadratic equation. While the factoring method is often the quickest for easily factorable equations, the quadratic formula provides a universal solution. Completing the square is a valuable technique that is also useful in other areas of mathematics, particularly in conic sections. Understanding all three methods allows you to choose the most efficient approach depending on the specific equation.

Applications of Quadratic Equations

Quadratic equations are far from merely abstract mathematical concepts. They have wide-ranging applications in various fields:

• 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 in materials.
• Economics: Modeling supply and demand, optimizing production, and analyzing market trends.
• Computer Graphics: Creating curves and shapes in computer-generated images and animations.

Frequently Asked Questions (FAQ)

Q: What if the quadratic equation doesn't factor easily?

A: In such cases, the quadratic formula is the most reliable method to find the solutions.

Q: Can a quadratic equation have only one solution?

A: Yes, a quadratic equation can have one solution (a repeated root) if the discriminant (b² - 4ac) is equal to zero.

Q: Can a quadratic equation have no real solutions?

A: Yes, if the discriminant (b² - 4ac) is negative, the quadratic equation will have no real solutions, but it will have complex solutions involving the imaginary unit i.

Q: What does the discriminant tell us?

A: The discriminant (b² - 4ac) determines the nature of the solutions: * If b² - 4ac > 0, there are two distinct real solutions. * If b² - 4ac = 0, there is one repeated real solution. * If b² - 4ac < 0, there are no real solutions (two complex solutions).

Conclusion

Solving the quadratic equation x² + 2x - 48 = 0, whether through factoring, the quadratic formula, or completing the square, ultimately leads to the same solutions: x = 6 and x = -8. Understanding these methods isn't just about solving this single equation; it's about mastering a fundamental algebraic skill with numerous applications in various fields. The ability to solve quadratic equations is a crucial stepping stone in your mathematical journey, opening doors to more advanced concepts and real-world problem-solving. Remember to practice these methods with different equations to build your confidence and proficiency. Don't hesitate to revisit this guide as needed to reinforce your understanding.