What Are The Solutions Of The Quadratic Equation 49x2 9

What Are the Solutions of the Quadratic Equation 49x² = 9? A Comprehensive Guide

The seemingly simple quadratic equation, 49x² = 9, offers a valuable opportunity to delve into fundamental algebraic concepts and problem-solving techniques. While the solution might appear straightforward at first glance, understanding the underlying principles and exploring various solution methods enriches mathematical comprehension. This article provides a comprehensive guide to solving this equation, explaining the process step-by-step and highlighting crucial algebraic concepts.

Understanding Quadratic Equations

Before tackling the specific equation, let's establish a foundational understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (typically 'x') is 2. The general form of a quadratic equation is:

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).

Our equation, 49x² = 9, is a simplified form of a quadratic equation. It lacks the 'bx' term (b=0), making it a simpler case to solve.

Solving 49x² = 9: Method 1 – Isolating the Variable

The most straightforward approach to solving 49x² = 9 involves isolating the variable 'x'. This method leverages basic algebraic manipulation:

1. Subtract 9 from both sides: This step ensures that the equation is set equal to zero, aligning it with the standard form of a quadratic equation. While not strictly necessary for this particular equation, it’s a good habit to develop for solving more complex quadratics. The equation becomes:

   49x² - 9 = 0

2. Factor the difference of squares: Notice that both 49x² and 9 are perfect squares. Specifically, 49x² = (7x)² and 9 = 3². This allows us to factor the equation using the difference of squares formula: a² - b² = (a + b)(a - b).

   (7x + 3)(7x - 3) = 0

3. Apply the Zero Product Property: The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x:

   7x + 3 = 0 or 7x - 3 = 0

4. Solve for x:

   • For 7x + 3 = 0: Subtract 3 from both sides, then divide by 7: x = -3/7
   • For 7x - 3 = 0: Add 3 to both sides, then divide by 7: x = 3/7

Therefore, the solutions to the quadratic equation 49x² = 9 are x = 3/7 and x = -3/7.

Solving 49x² = 9: Method 2 – Using the Square Root Property

This method directly utilizes the square root property, which states that if x² = k, then x = ±√k. This approach is particularly efficient when dealing with quadratic equations lacking the 'bx' term.

1. Divide both sides by 49: This isolates the x² term:

   x² = 9/49

2. Take the square root of both sides: Remember to account for both the positive and negative square roots:

   x = ±√(9/49)

3. Simplify: The square root of 9 is 3, and the square root of 49 is 7. Therefore:

   x = ±3/7

This yields the same solutions as Method 1: x = 3/7 and x = -3/7.

Graphical Representation and Interpretation

The solutions to the quadratic equation 49x² = 9 can be visualized graphically. The equation represents a parabola, a U-shaped curve. The x-intercepts of this parabola – the points where the parabola crosses the x-axis – correspond to the solutions of the equation. In this case, the parabola intersects the x-axis at x = 3/7 and x = -3/7.

The parabola opens upwards because the coefficient of the x² term (49) is positive. If the coefficient were negative, the parabola would open downwards.

Expanding the Understanding: More Complex Quadratic Equations

While 49x² = 9 provides a straightforward illustration of solving quadratic equations, many other quadratic equations are more complex, incorporating the 'bx' term. Several methods can be used to solve these more complex equations:

• Factoring: This involves expressing the quadratic expression as a product of two linear factors. It's often the most efficient method when the factors are easily identifiable.

• Quadratic Formula: The quadratic formula is a universal method for solving any quadratic equation. It's particularly useful when factoring is difficult or impossible. The formula is:

  x = [-b ± √(b² - 4ac)] / 2a

• Completing the Square: This method involves manipulating the equation to create a perfect square trinomial, which can then be easily factored.

Applications of Quadratic Equations

Quadratic equations have widespread applications in various fields, including:

• Physics: Calculating projectile motion, determining the trajectory of objects under the influence of gravity.

• Engineering: Designing structures, analyzing stresses and strains in materials.

• Economics: Modeling supply and demand, optimizing production.

• Computer Graphics: Creating curves and shapes in computer-generated images.

Further Exploration: Complex Numbers

While the equation 49x² = 9 yields real number solutions, some quadratic equations have solutions that are complex numbers. Complex numbers involve the imaginary unit 'i', where i² = -1. These solutions arise when the discriminant (b² - 4ac) in the quadratic formula is negative.

Conclusion: Mastering Quadratic Equations

Solving the equation 49x² = 9 provides a solid foundation for understanding quadratic equations. Mastering the various solution methods – isolating the variable, using the square root property, factoring, the quadratic formula, and completing the square – empowers you to tackle more complex quadratic problems. Understanding the graphical representation and the broader applications of quadratic equations further enhances your mathematical skills and problem-solving abilities. Remember that practice is key to developing proficiency in solving quadratic equations and applying them effectively in various contexts. By consistently working through different types of problems, you'll build confidence and a deeper comprehension of this essential algebraic concept.