Use The General Solution To Solve 5 6x 8x 17

Solving 5 + 6x + 8x² = 17 Using the General Solution for Quadratic Equations

This article will guide you through solving the quadratic equation 5 + 6x + 8x² = 17. Which means we'll explore the concept of quadratic equations, understand the general solution, and then apply it step-by-step to find the solutions for x. We'll also discuss the significance of the general solution and how it applies to a wide range of quadratic problems. Finally, we'll address frequently asked questions about solving quadratic equations.

Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually x) is 2. It's generally represented in the standard form:

ax² + bx + c = 0

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero (otherwise, it wouldn't be a quadratic equation). These values are called the roots or solutions of the equation. Solving a quadratic equation means finding the values of 'x' that satisfy the equation. Quadratic equations can have two, one, or zero real solutions, depending on the discriminant (a value we'll explore later).

Understanding the General Solution (Quadratic Formula)

The general solution, also known as the quadratic formula, provides a direct method for finding the roots of any quadratic equation. It's derived from completing the square method and offers a concise way to solve for x without needing to factor the equation (which isn't always possible or easy). The quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

This formula gives us two possible solutions for x, represented by the ± symbol (plus or minus). The term inside the square root, (b² - 4ac), is called the discriminant. The discriminant determines the nature of the roots:

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

Step-by-Step Solution of 5 + 6x + 8x² = 17

Before applying the quadratic formula, we need to rearrange our equation (5 + 6x + 8x² = 17) into the standard form (ax² + bx + c = 0):

1. Rearrange the Equation: Subtract 17 from both sides of the equation:

   8x² + 6x + 5 - 17 = 0

   This simplifies to:

   8x² + 6x - 12 = 0

Now we can identify our a, b, and c values:

• a = 8
• b = 6
• c = -12

2. Apply the Quadratic Formula: Substitute the values of a, b, and c into the quadratic formula:

   x = [-6 ± √(6² - 4 * 8 * -12)] / (2 * 8)

3. Simplify the Expression:

   x = [-6 ± √(36 + 384)] / 16

   x = [-6 ± √420] / 16

4. Simplify the Square Root: √420 can be simplified. Since 420 = 4 * 105, we can rewrite it as:

   x = [-6 ± 2√105] / 16

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5. Further Simplification: We can divide both the numerator and the denominator by 2:

   x = [-3 ± √105] / 8

Which means, the two solutions for x are:

• x = (-3 + √105) / 8
• x = (-3 - √105) / 8

These are the exact solutions. If you need approximate decimal values, you can use a calculator to find the approximate value of √105 (approximately 10.247) and then calculate the two solutions.

Significance of the General Solution

The general solution is incredibly powerful because it works for any quadratic equation, regardless of whether it's easily factorable or not. This makes it a fundamental tool in algebra and various fields that put to use quadratic models, such as physics, engineering, and economics. It provides a systematic and reliable method for finding solutions, eliminating the need for trial-and-error or other less efficient approaches.

Illustrative Examples and Applications

The quadratic formula finds applications in a wide array of real-world problems. For example:

• Projectile Motion: The trajectory of a projectile (like a ball thrown in the air) can be described by a quadratic equation. The quadratic formula helps determine the time it takes to reach a certain height or the maximum height it achieves.
• Area Calculations: Finding the dimensions of a rectangular area given its area and a relationship between its sides often involves solving a quadratic equation.
• Optimization Problems: Many optimization problems (finding maximum or minimum values) in various fields lead to quadratic equations that are solved using the quadratic formula.
• Engineering Design: In engineering design, quadratic equations appear when dealing with stress, strain, and deflection in structures.

Frequently Asked Questions (FAQ)

• Q: What if the discriminant is negative? A: If the discriminant (b² - 4ac) is negative, the quadratic equation has no real solutions. The solutions are complex numbers involving the imaginary unit i (where i² = -1).

• Q: Can I always use the quadratic formula? A: Yes, the quadratic formula is a universal method for solving quadratic equations. While factoring might be faster in some simple cases, the quadratic formula always guarantees a solution (real or complex).

• Q: What if 'a' is zero? A: If 'a' is zero, the equation is no longer a quadratic equation; it becomes a linear equation, which can be solved using simpler algebraic techniques.

• Q: How do I check my solutions? A: Substitute your calculated values of 'x' back into the original quadratic equation. If the equation holds true (both sides are equal), your solutions are correct.

Conclusion

Solving quadratic equations is a fundamental skill in mathematics. In practice, the general solution, or quadratic formula, provides a dependable and efficient method to find the roots of any quadratic equation, regardless of its complexity. Day to day, understanding the quadratic formula and its applications is essential for anyone studying mathematics, science, or engineering, as it underpins many important concepts and problem-solving techniques across various fields. Remember to always carefully rearrange the equation to the standard form before applying the formula to avoid mistakes. Mastering the quadratic formula empowers you to tackle numerous challenging problems and deepen your understanding of algebraic principles.