Solve The Equation $3x^2 + 17x - 6 = 0$.$\[ \begin{array}{r} 3x^2 + 18x - X - 6 = 0 \\ 3x(x + 6) - 1(x + 6) = 0 \\ (3x - 1)(x + 6) = 0 \\ 3x - 1 = 0 \text{ Or } X + 6 = 0 \end{array} \\]Based On The Work Shown Above, Which Of These

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving the quadratic equation $3x^2 + 17x - 6 = 0$ using the method of factoring. We will break down the solution into manageable steps, making it easy to understand and follow.

The Quadratic Equation

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $a$ is not equal to zero.

Solving the Equation

To solve the equation $3x^2 + 17x - 6 = 0$, we will use the method of factoring. This involves expressing the quadratic expression as a product of two binomials.

Step 1: Factor the Quadratic Expression

We start by factoring the quadratic expression $3x^2 + 17x - 6$. To do this, we need to find two numbers whose product is $3 \times (-6) = -18$ and whose sum is $17$. These numbers are $18$ and $-1$, so we can write:

$$3x^2 + 17x - 6 = 3x^2 + 18x - x - 6$$

Step 2: Factor by Grouping

Now, we can factor the expression by grouping:

$$3x^2 + 18x - x - 6 = 3x(x + 6) - 1(x + 6)$$

Step 3: Factor the Difference of Squares

We can now factor the difference of squares:

$$3x(x + 6) - 1(x + 6) = (3x - 1)(x + 6)$$

Step 4: Set Each Factor Equal to Zero

To find the solutions to the equation, we set each factor equal to zero:

$$(3x - 1)(x + 6) = 0$$

This gives us two possible solutions:

$$3x - 1 = 0 \text{ or } x + 6 = 0$$

Step 5: Solve for $x$

We can now solve for $x$ in each of the two equations:

$$3x - 1 = 0 \Rightarrow 3x = 1 \Rightarrow x = \frac{1}{3}$$

$$x + 6 = 0 \Rightarrow x = -6$$

Therefore, the solutions to the equation $3x^2 + 17x - 6 = 0$ are $x = \frac{1}{3}$ and $x = -6$.

Conclusion

Solving quadratic equations is an essential skill in mathematics, and factoring is a powerful method for solving these equations. By breaking down the solution into manageable steps, we can make the process easier to understand and follow. In this article, we solved the quadratic equation $3x^2 + 17x - 6 = 0$ using the method of factoring, and we found the solutions to be $x = \frac{1}{3}$ and $x = -6$.

Real-World Applications

Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model economic systems, including supply and demand curves, and to optimize business decisions.

Tips and Tricks

Here are some tips and tricks for solving quadratic equations:

• Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. It is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Check for factoring: Before using the quadratic formula, check if the quadratic expression can be factored. If it can, use factoring to solve the equation.
• Use technology: Quadratic equations can be solved using technology, such as graphing calculators and computer software. These tools can help you visualize the solutions and make it easier to find them.

Practice Problems

Here are some practice problems to help you reinforce your understanding of solving quadratic equations:

• Problem 1: Solve the quadratic equation $x^2 + 5x + 6 = 0$ using factoring.
• Problem 2: Solve the quadratic equation $2x^2 - 7x - 3 = 0$ using the quadratic formula.
• Problem 3: Solve the quadratic equation $x^2 + 2x - 5 = 0$ using factoring.

Conclusion

$$$$$$

Frequently Asked Questions

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $a$ is not equal to zero.

Q: How do I solve a quadratic equation?

A: There are several methods for solving quadratic equations, including:

• Factoring: This involves expressing the quadratic expression as a product of two binomials.
• Quadratic formula: This is a powerful tool for solving quadratic equations. It is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Graphing: This involves graphing the quadratic function and finding the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: When should I use the quadratic formula?

A: You should use the quadratic formula when:

• Factoring is not possible: If the quadratic expression cannot be factored, the quadratic formula is a good alternative.
• The equation is complex: If the equation is complex, the quadratic formula can help simplify the solution.
• You need a precise solution: If you need a precise solution, the quadratic formula is a good choice.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. The general form of a linear equation is:

$$ax + b = 0$$

where $a$ and $b$ are constants, and $a$ is not equal to zero.

Q: Can I use the quadratic formula to solve a linear equation?

A: No, you cannot use the quadratic formula to solve a linear equation. The quadratic formula is specifically designed for solving quadratic equations.

Q: What is the relationship between quadratic equations and quadratic functions?

A: Quadratic equations and quadratic functions are closely related. A quadratic function is a function of the form:

$$f(x) = ax^2 + bx + c$$

where $a$, $b$, and $c$ are constants, and $a$ is not equal to zero. Quadratic equations are used to find the x-intercepts of quadratic functions.

Q: Can I use technology to solve quadratic equations?

A: Yes, you can use technology to solve quadratic equations. Graphing calculators and computer software can help you visualize the solutions and make it easier to find them.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model economic systems, including supply and demand curves, and to optimize business decisions.

Q: Can I use quadratic equations to solve systems of equations?

A: Yes, you can use quadratic equations to solve systems of equations. By solving the quadratic equation, you can find the x-intercepts of the quadratic function, which can help you solve the system of equations.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

• Not checking for factoring: Make sure to check if the quadratic expression can be factored before using the quadratic formula.
• Not using the correct formula: Make sure to use the correct formula for the quadratic equation, including the correct values for $a$, $b$, and $c$.
• Not checking for complex solutions: Make sure to check for complex solutions, including complex numbers and imaginary numbers.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding the basics of quadratic equations, including the quadratic formula and factoring, you can solve a wide range of problems and applications. In this article, we answered some frequently asked questions about quadratic equations, including their definition, solution methods, and real-world applications. We also discussed some common mistakes to avoid when solving quadratic equations.