What Are The Solutions Of 3x2 14x 16 0

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

This article delves into the various methods for solving the quadratic equation 3x² + 14x + 16 = 0. We'll explore different approaches, emphasizing their practical applications and highlighting the strengths and weaknesses of each method. Understanding quadratic equations is fundamental in algebra and has widespread applications in various fields, from physics and engineering to economics and computer science.

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. In our specific case, a = 3, b = 14, and c = 16.

Method 1: Factoring

Factoring is a direct method that involves expressing the quadratic equation as a product of two linear expressions. This method is efficient when the factors are easily identifiable. Let's attempt to factor 3x² + 14x + 16 = 0:

We are looking for two numbers that add up to 14 (the coefficient of x) and multiply to 48 (the product of a and c, which is 3 * 16). These numbers are 8 and 6. Therefore, we can rewrite the equation as:

3x² + 8x + 6x + 16 = 0

Now, we can factor by grouping:

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

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

This gives us two possible solutions:

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

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

Advantages of Factoring:

• Simple and elegant: When the equation is easily factorable, this is the quickest method.
• Provides insight: Factoring gives a clear understanding of the roots and their relationship to the original equation.

Disadvantages of Factoring:

• Not always possible: Many quadratic equations are not easily factorable using integers.
• Can be time-consuming: Finding the right factors can be challenging and trial-and-error intensive.

Method 2: Quadratic Formula

The quadratic formula is a universal method that works for all quadratic equations, regardless of whether they are easily factorable. The formula is derived from completing the square and provides a direct way to calculate the roots:

x = [-b ± √(b² - 4ac)] / 2a

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

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

The solutions using the quadratic formula are the same as those obtained through factoring: x = -2 and x = -8/3.

Advantages of the Quadratic Formula:

• Universally applicable: Works for all quadratic equations, regardless of their factorability.
• Direct calculation: Provides a straightforward method to calculate the roots.

Disadvantages of the Quadratic Formula:

• Can be more complex: Involves more calculation steps compared to factoring.
• Potential for errors: Errors can occur if calculations are not performed carefully.

Method 3: Completing the Square

Completing the square is another method used to solve quadratic equations. It involves manipulating the equation to form a perfect square trinomial, which can then be easily factored. Let's apply this method to our equation:

3x² + 14x + 16 = 0

First, divide the equation by the coefficient of x² (a = 3):

x² + (14/3)x + (16/3) = 0

Move the constant term to the right side of the equation:

x² + (14/3)x = -16/3

Now, we need to complete the square. 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 of the equation:

x² + (14/3)x + 49/9 = -16/3 + 49/9

The left side is now a perfect square trinomial:

(x + 7/3)² = -48/9 + 49/9 = 1/9

Take the square root of both sides:

x + 7/3 = ±√(1/9) = ±1/3

Solve for x:

x = -7/3 ± 1/3

This leads to the same solutions as before:

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

Advantages of Completing the Square:

• Illustrates a fundamental algebraic concept: Provides a deeper understanding of the relationship between quadratic equations and perfect squares.
• Useful in other areas of mathematics: This technique has broader applications beyond solving quadratic equations.

Disadvantages of Completing the Square:

• Can be cumbersome: Involves more steps compared to the quadratic formula, especially with fractional coefficients.
• Prone to errors: Requires careful attention to detail, and mistakes can easily occur.

Graphical Representation and Interpretation

The solutions of the quadratic equation represent the x-intercepts (or roots) of the parabola representing the function y = 3x² + 14x + 16. Graphing the quadratic function reveals that the parabola intersects the x-axis at x = -2 and x = -8/3, confirming our calculated solutions. This graphical representation provides a visual understanding of the solutions.

Applications of Quadratic Equations

Quadratic equations have numerous applications across various fields:

• Physics: Calculating projectile motion, analyzing the trajectory of objects under gravity.
• Engineering: Designing structures, optimizing shapes and sizes for maximum strength and stability.
• Economics: Modeling supply and demand curves, optimizing production costs.
• Computer Science: Developing algorithms and simulations.

Conclusion

Solving the quadratic equation 3x² + 14x + 16 = 0 can be achieved through various methods: factoring, the quadratic formula, and completing the square. The choice of method depends on the specific equation and individual preferences. While factoring is elegant and efficient for simple equations, the quadratic formula offers a universal solution applicable to all quadratic equations. Completing the square provides a deeper understanding of the underlying algebraic principles. Understanding these methods empowers you to tackle a wide range of problems involving quadratic equations, opening doors to more advanced mathematical concepts and their real-world applications. Regardless of the method used, the solutions for the given equation remain consistent: x = -2 and x = -8/3. Remember to always check your solutions by substituting them back into the original equation to verify their accuracy.