Solve The Following Equations: A) X^2 - 50 = 0 B) 2x^2 - 7 = 0 C) 3x^2 - 24 = 0 D) 28x^2 - 14 = 0 E) 27x^2 - 18 = 0 F) 9x^2 + 4 = 0

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In the realm of mathematics, quadratic equations hold a significant position, often appearing in various contexts, from physics to engineering. Mastering the techniques to solve these equations is crucial for any aspiring mathematician or scientist. In this article, we will delve into the methods for solving quadratic equations of a specific form: ax² + c = 0. We will dissect each equation step-by-step, providing clear explanations and insights into the underlying principles. This comprehensive guide aims to equip you with the necessary skills to confidently tackle similar problems and deepen your understanding of quadratic equations.

Understanding Quadratic Equations

Before we embark on solving the equations, let's first establish a firm grasp of what quadratic equations are. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. However, the equations we will be tackling in this article are of a simpler form, ax² + c = 0, where the coefficient of the linear term (b) is zero. This simplification allows us to employ a more direct approach to finding the solutions.

The solutions to a quadratic equation are the values of the variable (usually denoted as x) that satisfy the equation, meaning when these values are substituted into the equation, the left-hand side equals the right-hand side. These solutions are also known as the roots or zeros of the quadratic equation. A quadratic equation can have up to two distinct real solutions, one real solution (a repeated root), or no real solutions, depending on the discriminant (b² - 4ac) of the general quadratic equation. In the case of our simplified form (ax² + c = 0), the nature of the solutions depends on the signs of the coefficients a and c.

Methods for Solving ax² + c = 0

The primary method for solving quadratic equations of the form ax² + c = 0 involves isolating the x² term and then taking the square root of both sides. This approach leverages the fundamental properties of algebraic manipulation and the definition of square roots. Let's outline the general steps involved:

  1. Isolate the x² term: This involves performing algebraic operations to get the x² term by itself on one side of the equation. Typically, this involves adding or subtracting the constant term (c) from both sides and then dividing both sides by the coefficient of the x² term (a).
  2. Take the square root of both sides: Once the x² term is isolated, we take the square root of both sides of the equation. It's crucial to remember that taking the square root introduces both positive and negative solutions, as both the positive and negative square roots of a number, when squared, result in the original number.
  3. Solve for x: After taking the square root, we obtain two possible values for x. These values represent the solutions to the quadratic equation.

Solving the Equations

Now, let's apply this method to solve the given equations:

a) x² - 50 = 0

Our goal is to find the values of x that satisfy this equation. Following the steps outlined above:

  1. Isolate the x² term: Add 50 to both sides of the equation: x² = 50
  2. Take the square root of both sides: √x² = ±√50 x = ±√50
  3. Simplify the radical: We can simplify √50 by factoring out the largest perfect square, which is 25: x = ±√(25 * 2) x = ±√25 * √2 x = ±5√2

Therefore, the solutions to the equation x² - 50 = 0 are x = 5√2 and x = -5√2.

b) 2x² - 7 = 0

  1. Isolate the x² term: Add 7 to both sides: 2x² = 7 Divide both sides by 2: x² = 7/2
  2. Take the square root of both sides: √x² = ±√(7/2) x = ±√(7/2)
  3. Rationalize the denominator (optional): To rationalize the denominator, we multiply both the numerator and denominator by √2: x = ±(√7 / √2) * (√2 / √2) x = ±(√14 / 2)

Thus, the solutions to the equation 2x² - 7 = 0 are x = √14 / 2 and x = -√14 / 2.

c) 3x² - 24 = 0

  1. Isolate the x² term: Add 24 to both sides: 3x² = 24 Divide both sides by 3: x² = 8
  2. Take the square root of both sides: √x² = ±√8 x = ±√8
  3. Simplify the radical: x = ±√(4 * 2) x = ±√4 * √2 x = ±2√2

Hence, the solutions to the equation 3x² - 24 = 0 are x = 2√2 and x = -2√2.

d) 28x² - 14 = 0

  1. Isolate the x² term: Add 14 to both sides: 28x² = 14 Divide both sides by 28: x² = 14/28 Simplify the fraction: x² = 1/2
  2. Take the square root of both sides: √x² = ±√(1/2) x = ±√(1/2)
  3. Rationalize the denominator (optional): x = ±(√1 / √2) * (√2 / √2) x = ±(√2 / 2)

Therefore, the solutions to the equation 28x² - 14 = 0 are x = √2 / 2 and x = -√2 / 2.

e) 27x² - 18 = 0

  1. Isolate the x² term: Add 18 to both sides: 27x² = 18 Divide both sides by 27: x² = 18/27 Simplify the fraction: x² = 2/3
  2. Take the square root of both sides: √x² = ±√(2/3) x = ±√(2/3)
  3. Rationalize the denominator (optional): x = ±(√2 / √3) * (√3 / √3) x = ±(√6 / 3)

Thus, the solutions to the equation 27x² - 18 = 0 are x = √6 / 3 and x = -√6 / 3.

f) 9x² + 4 = 0

  1. Isolate the x² term: Subtract 4 from both sides: 9x² = -4 Divide both sides by 9: x² = -4/9
  2. Take the square root of both sides: √x² = ±√(-4/9) x = ±√(-4/9)

In this case, we encounter a negative value under the square root. This indicates that there are no real solutions to this equation. The solutions would be complex numbers, which involve the imaginary unit 'i', where i² = -1. Therefore, the equation 9x² + 4 = 0 has no real solutions.

Key Takeaways

Solving quadratic equations of the form ax² + c = 0 involves isolating the x² term, taking the square root of both sides, and simplifying the resulting expressions. Remember to consider both positive and negative square roots. When encountering a negative value under the square root, the equation has no real solutions. This step-by-step approach provides a systematic way to solve these equations with confidence.

Understanding the nature of solutions is crucial in mathematics. In the equations we solved, we observed that the solutions can be real numbers (rational or irrational) or, in some cases, there might be no real solutions. This understanding is fundamental in various mathematical applications and problem-solving scenarios.

Conclusion

This article has provided a comprehensive guide to solving quadratic equations of the form ax² + c = 0. By understanding the underlying principles and following the outlined steps, you can confidently tackle these equations. Remember to practice regularly and apply these techniques to various problems to solidify your understanding. Mastering quadratic equations is a significant step in your mathematical journey, opening doors to more advanced concepts and applications. The ability to solve equations effectively is a cornerstone of mathematical proficiency, and this guide has hopefully equipped you with the tools to do just that.