What Are The Solutions To The Quadratic Equation 4x2 64

What Are the Solutions to the Quadratic Equation 4x² = 64? A Comprehensive Guide

Solving quadratic equations is a fundamental skill in algebra, crucial for numerous applications in mathematics, physics, engineering, and computer science. This article delves into the solution of the specific quadratic equation 4x² = 64, exploring multiple methods, explaining the underlying concepts, and demonstrating how to verify the solutions obtained. We will also discuss the broader context of quadratic equations and their significance.

Understanding Quadratic Equations

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' is not equal to zero (otherwise, it wouldn't be a quadratic equation). Our equation, 4x² = 64, is a simplified form of a quadratic equation where b = 0 and c = -64.

Method 1: Isolating the Variable

The simplest approach to solving 4x² = 64 is to isolate the variable 'x'. We can achieve this through a series of algebraic manipulations:

1. Divide both sides by 4: This simplifies the equation to x² = 16.

2. Take the square root of both sides: Remember that the square root of a number can be positive or negative. Therefore, we get two solutions:

   x = +√16 = 4 x = -√16 = -4

Therefore, the solutions to the equation 4x² = 64 are x = 4 and x = -4.

Method 2: Factoring

Factoring is another common method for solving quadratic equations. While this method is particularly useful when 'b' is not equal to zero, we can still adapt it for our equation:

1. Rewrite the equation: First, rewrite the equation in the standard form ax² + bx + c = 0:

   4x² - 64 = 0

2. Factor out the greatest common factor (GCF): The GCF of 4x² and -64 is 4. Factoring it out gives:

   4(x² - 16) = 0

3. Factor the difference of squares: The expression (x² - 16) is a difference of squares (a² - b² = (a + b)(a - b), where a = x and b = 4). Factoring it gives:

   4(x + 4)(x - 4) = 0

4. Set each factor to zero: This results in three equations:

   4 = 0 (This is always false, so we can ignore it) x + 4 = 0 => x = -4 x - 4 = 0 => x = 4

Again, the solutions are x = 4 and x = -4.

Method 3: Quadratic Formula

The quadratic formula is a powerful tool that solves any quadratic equation, regardless of the values of 'a', 'b', and 'c'. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

In our case, a = 4, b = 0, and c = -64. Substituting these values into the quadratic formula:

x = [0 ± √(0² - 4 * 4 * -64)] / (2 * 4) x = [± √(1024)] / 8 x = ± 32 / 8 x = ± 4

Once again, the solutions are x = 4 and x = -4.

Verifying the Solutions

It's crucial to verify the solutions by substituting them back into the original equation:

For x = 4:

4(4)² = 4 * 16 = 64. This is correct.

For x = -4:

4(-4)² = 4 * 16 = 64. This is also correct.

The Significance of Quadratic Equations

Quadratic equations are far more than just abstract algebraic exercises. They have wide-ranging applications across various fields:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity, and analyzing oscillatory systems.
• Engineering: Designing structures, analyzing stresses and strains in materials, and optimizing various engineering processes.
• Computer Graphics: Creating curves and surfaces in 3D modeling and animation.
• Economics: Modeling cost functions, revenue functions, and profit maximization.
• Finance: Calculating compound interest and determining investment growth.

Solving More Complex Quadratic Equations

While the equation 4x² = 64 is relatively straightforward, the principles and methods discussed here apply to more complex quadratic equations as well. Remember to:

• Simplify the equation: Combine like terms and rearrange the equation into the standard form.
• Choose the appropriate method: Factoring is efficient for simpler equations, while the quadratic formula works for all cases.
• Always verify your solutions: Substitute the solutions back into the original equation to ensure accuracy.

Dealing with Complex Roots

Some quadratic equations have no real solutions. This occurs when the discriminant (b² - 4ac) is negative, resulting in imaginary or complex roots. These roots involve the imaginary unit 'i', where i² = -1. Understanding complex numbers is essential for tackling these types of equations.

Graphical Representation

Quadratic equations represent parabolas when graphed. The solutions to the equation are the x-intercepts (where the parabola intersects the x-axis). Visualizing the graph can provide valuable insights into the nature and number of solutions.

Conclusion

Solving the quadratic equation 4x² = 64 demonstrates the fundamental principles of solving quadratic equations. We explored three different methods – isolating the variable, factoring, and the quadratic formula – all leading to the same solutions: x = 4 and x = -4. The importance of verifying solutions and the broader significance of quadratic equations in various fields were also highlighted. By mastering these concepts and techniques, you will gain a strong foundation in algebra and its applications. Remember to practice regularly and explore more complex examples to enhance your understanding and problem-solving skills.