Solving Quadratic Equations 9x² - 64 = 0 A Comprehensive Guide
In the realm of mathematics, quadratic equations hold a significant position, often encountered in various fields, from physics to engineering. These equations, characterized by their highest power of the variable being two, present a unique challenge in finding their solutions, also known as roots or zeros. This article delves into the intricacies of solving a specific quadratic equation, 9x² - 64 = 0, meticulously examining the steps involved and identifying the correct solutions from a given set of options.
Deciphering the Quadratic Equation: 9x² - 64 = 0
Our primary objective is to determine the values of 'x' that satisfy the equation 9x² - 64 = 0. This equation, a quintessential example of a quadratic equation, can be solved using various methods, including factoring, completing the square, or employing the quadratic formula. In this instance, we'll leverage the factoring method, which proves to be particularly efficient for equations exhibiting a specific structure.
The equation 9x² - 64 = 0 presents itself as a difference of squares, a pattern that allows for straightforward factorization. Recognizing this pattern is crucial, as it simplifies the solution process significantly. The difference of squares pattern states that a² - b² can be factored into (a + b)(a - b). In our equation, 9x² corresponds to a², where a = 3x, and 64 corresponds to b², where b = 8. Applying the difference of squares pattern, we can rewrite the equation as:
(3x + 8)(3x - 8) = 0
This factored form sets the stage for finding the solutions. The principle behind this step lies in the zero-product property, which asserts that if the product of two factors equals zero, then at least one of the factors must be zero. Consequently, we can set each factor equal to zero and solve for 'x'.
Solving for x: Unraveling the Roots
To find the solutions, we set each factor equal to zero:
3x + 8 = 0
and
3x - 8 = 0
Solving the first equation, 3x + 8 = 0, involves isolating 'x'. Subtracting 8 from both sides yields:
3x = -8
Dividing both sides by 3, we obtain:
x = -8/3
This value, -8/3, represents one of the solutions to the quadratic equation. It signifies a point where the quadratic function intersects the x-axis.
Turning our attention to the second equation, 3x - 8 = 0, we again isolate 'x'. Adding 8 to both sides results in:
3x = 8
Dividing both sides by 3, we find:
x = 8/3
This value, 8/3, constitutes the second solution to the quadratic equation. It mirrors the first solution in magnitude but possesses the opposite sign, reflecting the symmetrical nature of quadratic equations.
Therefore, the solutions to the equation 9x² - 64 = 0 are x = -8/3 and x = 8/3. These values are the roots of the quadratic equation, the points where the parabola intersects the x-axis.
Evaluating the Options: Identifying the Correct Solutions
Now that we have determined the solutions to the equation, we can evaluate the provided options to identify the correct ones. The options are:
A. -8 B. -3/8 C. 8/3 D. -8/3 E. 3/8 F. 8
Comparing these options with our calculated solutions, we can readily identify the correct answers:
- Option C, 8/3, matches one of our solutions.
- Option D, -8/3, aligns with the other solution we found.
Therefore, the correct solutions to the equation 9x² - 64 = 0 are C. 8/3 and D. -8/3. These values, when substituted back into the original equation, will satisfy the equality, confirming their validity as solutions.
The remaining options, A, B, E, and F, do not correspond to the solutions we calculated. Substituting these values into the original equation would not result in a true statement, indicating that they are not solutions.
Alternative Methods for Solving Quadratic Equations
While factoring proved to be an efficient method for solving the equation 9x² - 64 = 0, it's essential to acknowledge that other methods exist, each with its own strengths and applicability. Two prominent alternative methods are the quadratic formula and completing the square.
The Quadratic Formula: A Universal Solution
The quadratic formula stands as a universal tool for solving any quadratic equation, regardless of its factorability. It provides a direct pathway to the solutions, circumventing the need for factoring or manipulating the equation into a specific form. The quadratic formula is expressed as:
x = (-b ± √(b² - 4ac)) / 2a
where 'a', 'b', and 'c' are the coefficients of the quadratic equation in the standard form ax² + bx + c = 0. In our equation, 9x² - 64 = 0, we have a = 9, b = 0, and c = -64. Substituting these values into the quadratic formula, we get:
x = (0 ± √(0² - 4 * 9 * -64)) / (2 * 9)
Simplifying the expression under the square root:
x = (0 ± √(2304)) / 18
x = (0 ± 48) / 18
This yields two solutions:
x = 48 / 18 = 8/3
and
x = -48 / 18 = -8/3
These solutions align perfectly with those obtained through factoring, demonstrating the quadratic formula's efficacy in solving quadratic equations.
Completing the Square: Transforming the Equation
Completing the square is another powerful technique for solving quadratic equations. It involves manipulating the equation into a perfect square trinomial, which can then be easily solved by taking the square root. To apply this method to 9x² - 64 = 0, we first isolate the x² term:
9x² = 64
Dividing both sides by 9:
x² = 64/9
Now, we take the square root of both sides:
x = ±√(64/9)
x = ±8/3
Again, we arrive at the same solutions, x = 8/3 and x = -8/3, reinforcing the consistency of different solution methods.
The Significance of Quadratic Equations
Quadratic equations are not merely abstract mathematical constructs; they possess profound significance in various real-world applications. Their ability to model parabolic trajectories makes them indispensable in fields such as physics, where they describe the motion of projectiles under gravity. In engineering, quadratic equations play a crucial role in designing structures, optimizing processes, and analyzing systems.
The solutions to quadratic equations, the roots or zeros, often represent critical points in the context of a problem. For instance, in projectile motion, the roots might correspond to the launch and landing points of a projectile. In optimization problems, they might indicate the points of maximum or minimum value.
Conclusion: Mastering Quadratic Equations
In conclusion, solving quadratic equations is a fundamental skill in mathematics with far-reaching implications. The equation 9x² - 64 = 0, a prime example of a difference of squares, can be efficiently solved through factoring, yielding the solutions x = 8/3 and x = -8/3. These solutions, confirmed by the quadratic formula and completing the square, highlight the versatility of different solution methods.
Understanding quadratic equations and mastering the techniques for solving them empowers us to tackle a wide array of problems in mathematics, science, and engineering. From modeling projectile motion to designing optimal structures, quadratic equations serve as indispensable tools for understanding and shaping the world around us. By delving into the intricacies of quadratic equations, we unlock a deeper appreciation for the power and beauty of mathematics.
