What Are The Solutions Of 3x2 14x 16 0

Solving the Quadratic Equation: 3x² + 14x + 16 = 0

This article delves into the complete solution of the quadratic equation 3x² + 14x + 16 = 0, exploring various methods and providing a comprehensive understanding of the process. Quadratic equations are fundamental in algebra and have wide-ranging applications in various fields, including physics, engineering, and computer science. Understanding how to solve them is crucial for anyone studying mathematics or related disciplines.

Understanding Quadratic Equations

A quadratic equation is an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The highest power of the variable 'x' is 2, which gives the equation its quadratic nature. The solutions to a quadratic equation are the values of 'x' that satisfy the equation. These solutions are also known as roots or zeros.

Our specific equation, 3x² + 14x + 16 = 0, fits this general form, with a = 3, b = 14, and c = 16. We will explore several methods to find the values of 'x' that make this equation true.

Method 1: Factoring

Factoring is a method used to find the solutions of a quadratic equation by expressing the quadratic expression as a product of two linear expressions. This method is particularly useful when the quadratic expression can be easily factored.

To factor 3x² + 14x + 16, we look for two numbers that add up to 14 (the coefficient of x) and multiply to 48 (the product of the coefficient of x² and the constant term, 3 * 16). These two numbers are 6 and 8. We can rewrite the equation as:

3x² + 6x + 8x + 16 = 0

Now, we can factor by grouping:

3x(x + 2) + 8(x + 2) = 0

This simplifies to:

(3x + 8)(x + 2) = 0

This factored form tells us that the equation is satisfied if either (3x + 8) = 0 or (x + 2) = 0. Solving these linear equations gives us the solutions:

• 3x + 8 = 0 => 3x = -8 => x = -8/3
• x + 2 = 0 => x = -2

Therefore, the solutions to the equation 3x² + 14x + 16 = 0 using factoring are x = -8/3 and x = -2.

Method 2: Quadratic Formula

The quadratic formula provides a general solution for any quadratic equation. It's a powerful tool that works even when factoring is difficult or impossible. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Plugging in the values from our equation (a = 3, b = 14, c = 16), we get:

x = [-14 ± √(14² - 4 * 3 * 16)] / (2 * 3)

x = [-14 ± √(196 - 192)] / 6

x = [-14 ± √4] / 6

x = [-14 ± 2] / 6

This gives us two solutions:

• x = (-14 + 2) / 6 = -12 / 6 = -2
• x = (-14 - 2) / 6 = -16 / 6 = -8/3

Again, we find the same solutions: x = -2 and x = -8/3.

Method 3: Completing the Square

Completing the square is another method for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily factored.

Starting with 3x² + 14x + 16 = 0, we first divide the equation by the coefficient of x² (3):

x² + (14/3)x + (16/3) = 0

Next, we move the constant term to the right side:

x² + (14/3)x = -16/3

Now, we complete the square. We take half of the coefficient of x ((14/3)/2 = 7/3), square it ((7/3)² = 49/9), and add it to both sides:

x² + (14/3)x + 49/9 = -16/3 + 49/9

This gives us:

(x + 7/3)² = -48/9 + 49/9 = 1/9

Taking the square root of both sides:

x + 7/3 = ±√(1/9) = ±1/3

Solving for x:

• x = -7/3 + 1/3 = -6/3 = -2
• x = -7/3 - 1/3 = -8/3

Once again, we arrive at the same solutions: x = -2 and x = -8/3.

Verification of Solutions

To verify our solutions, we can substitute each value of x back into the original equation:

For x = -2:

3(-2)² + 14(-2) + 16 = 12 - 28 + 16 = 0

For x = -8/3:

3(-8/3)² + 14(-8/3) + 16 = 3(64/9) - 112/3 + 16 = 64/3 - 112/3 + 48/3 = 0

Both solutions satisfy the equation, confirming their correctness.

Graphical Representation

The solutions of a quadratic equation also represent the x-intercepts (points where the graph crosses the x-axis) of the parabola defined by the equation y = 3x² + 14x + 16. Graphing this parabola would show that it intersects the x-axis at x = -2 and x = -8/3. This visual representation provides further confirmation of our solutions.

Applications of Quadratic Equations

Quadratic equations have numerous applications in various fields. Some examples include:

• Physics: Calculating the trajectory of projectiles.
• Engineering: Designing bridges and structures.
• Computer Science: Developing algorithms and solving optimization problems.
• Economics: Modeling supply and demand curves.
• Finance: Determining the optimal investment strategies.

Conclusion

This comprehensive analysis demonstrates three different methods for solving the quadratic equation 3x² + 14x + 16 = 0, all yielding the same solutions: x = -2 and x = -8/3. Understanding these methods and their applications is crucial for success in mathematics and related fields. The choice of method often depends on the specific equation and personal preference, but the quadratic formula always provides a reliable solution. Remember to always verify your solutions by substituting them back into the original equation. The ability to solve quadratic equations efficiently is a foundational skill in mathematics and has wide-ranging implications across numerous disciplines.