Solving 7x^2 - 42x = 0 A Comprehensive Guide

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In the realm of mathematics, quadratic equations hold a prominent position, serving as fundamental tools for modeling and solving a myriad of real-world problems. From the trajectory of a projectile to the optimization of engineering designs, quadratic equations find applications across diverse fields. This comprehensive guide delves into the intricacies of solving the quadratic equation 7x^2 - 42x = 0, providing a step-by-step approach to finding its solutions and verifying their correctness. We will explore various techniques, including factoring, the quadratic formula, and graphical methods, to equip you with a versatile toolkit for tackling quadratic equations.

Understanding Quadratic Equations

At its core, a quadratic equation is a polynomial equation of the second degree, characterized by the general form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to a quadratic equation, also known as roots or zeros, represent the values of x that satisfy the equation. These solutions correspond to the points where the parabola defined by the quadratic equation intersects the x-axis. A quadratic equation can have two distinct real roots, one real root (a repeated root), or two complex roots, depending on the discriminant (b^2 - 4ac).

In our specific case, we are presented with the quadratic equation 7x^2 - 42x = 0. This equation is a special case where the constant term c is equal to zero. Such equations can be readily solved by factoring out the common factor x, simplifying the process of finding the solutions.

Factoring Method A Straightforward Approach

One of the most efficient methods for solving quadratic equations, especially those with a common factor, is the factoring method. This technique involves expressing the quadratic equation as a product of two linear factors. By setting each factor equal to zero, we can obtain the solutions to the equation.

To apply the factoring method to 7x^2 - 42x = 0, we first identify the common factor, which is 7x. Factoring out 7x from both terms, we get:

7x(x - 6) = 0

Now, we have expressed the quadratic equation as a product of two factors: 7x and (x - 6). To find the solutions, we set each factor equal to zero:

7x = 0 or x - 6 = 0

Solving the first equation, 7x = 0, we divide both sides by 7 to obtain x = 0.

Solving the second equation, x - 6 = 0, we add 6 to both sides to obtain x = 6.

Therefore, the solutions to the quadratic equation 7x^2 - 42x = 0 are x = 0 and x = 6. These solutions correspond to options D and E in the given choices.

Verification of Solutions Ensuring Accuracy

To ensure the accuracy of our solutions, we must verify that they satisfy the original equation. This involves substituting each solution back into the equation and confirming that the equation holds true.

Let's first verify x = 0:

7(0)^2 - 42(0) = 0

0 - 0 = 0

0 = 0

The equation holds true for x = 0, confirming that it is a valid solution.

Now, let's verify x = 6:

7(6)^2 - 42(6) = 0

7(36) - 252 = 0

252 - 252 = 0

0 = 0

The equation also holds true for x = 6, further solidifying its validity as a solution.

The Quadratic Formula A Universal Solution

While the factoring method is effective for certain quadratic equations, it may not always be applicable. For more complex equations or those that cannot be easily factored, the quadratic formula provides a universal solution. The quadratic formula is derived from the general form of a quadratic equation, ax^2 + bx + c = 0, and expresses the solutions in terms of the coefficients a, b, and c:

x = (-b ± √(b^2 - 4ac)) / 2a

To apply the quadratic formula to our equation, 7x^2 - 42x = 0, we identify the coefficients: a = 7, b = -42, and c = 0. Substituting these values into the quadratic formula, we get:

x = (42 ± √((-42)^2 - 4 * 7 * 0)) / (2 * 7)

x = (42 ± √(1764)) / 14

x = (42 ± 42) / 14

This yields two possible solutions:

x = (42 + 42) / 14 = 84 / 14 = 6

x = (42 - 42) / 14 = 0 / 14 = 0

As we can see, the quadratic formula confirms the solutions we obtained using the factoring method: x = 0 and x = 6.

Graphical Interpretation Visualizing the Solutions

Quadratic equations can also be interpreted graphically. The graph of a quadratic equation is a parabola, a U-shaped curve. The solutions to the equation correspond to the points where the parabola intersects the x-axis. These points are also known as the x-intercepts or roots of the equation.

To visualize the solutions to 7x^2 - 42x = 0, we can plot the graph of the equation y = 7x^2 - 42x. The parabola intersects the x-axis at two points: x = 0 and x = 6, which are the solutions we obtained algebraically.

The graphical interpretation provides a visual confirmation of our solutions, reinforcing their accuracy and providing a deeper understanding of the relationship between quadratic equations and their graphical representations.

Conclusion Mastering Quadratic Equations

In this comprehensive guide, we have explored the intricacies of solving the quadratic equation 7x^2 - 42x = 0, employing various techniques such as factoring, the quadratic formula, and graphical methods. We have demonstrated that the solutions to this equation are x = 0 and x = 6, corresponding to options D and E in the given choices. By understanding these methods and practicing their application, you can confidently tackle a wide range of quadratic equations and unlock their potential in various mathematical and real-world scenarios.

Key Takeaways:

  • Quadratic equations are polynomial equations of the second degree, with the general form ax^2 + bx + c = 0.
  • The solutions to a quadratic equation are the values of x that satisfy the equation, also known as roots or zeros.
  • The factoring method involves expressing the quadratic equation as a product of linear factors and setting each factor equal to zero.
  • The quadratic formula provides a universal solution for quadratic equations, expressed as x = (-b ± √(b^2 - 4ac)) / 2a.
  • The graphical interpretation of a quadratic equation is a parabola, where the solutions correspond to the x-intercepts.
  • Verifying solutions by substituting them back into the original equation ensures accuracy.

By mastering these concepts and techniques, you will be well-equipped to solve quadratic equations and apply them to various mathematical and real-world problems. Remember to practice consistently and explore different types of equations to enhance your understanding and problem-solving skills.

#Additional Tips for Solving Quadratic Equations

When solving quadratic equations, consider these additional tips to enhance your understanding and problem-solving skills:

  1. Simplify the equation: Before attempting to solve a quadratic equation, simplify it as much as possible. This may involve combining like terms, factoring out common factors, or rearranging the equation into the standard form (ax^2 + bx + c = 0). Simplifying the equation can make it easier to apply the factoring method or the quadratic formula.

  2. Check for special cases: Be aware of special cases of quadratic equations, such as those with a missing constant term (c = 0) or those that are perfect square trinomials. These special cases can often be solved more efficiently using specific techniques.

  3. Use the discriminant: The discriminant (b^2 - 4ac) provides valuable information about the nature of the solutions. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root (a repeated root). If the discriminant is negative, the equation has two complex roots.

  4. Estimate the solutions: Before applying any solving method, try to estimate the solutions. This can help you check the reasonableness of your answers and identify potential errors.

  5. Practice regularly: Consistent practice is key to mastering quadratic equations. Solve a variety of problems, including those with different coefficients and forms, to develop your skills and intuition.

By incorporating these tips into your problem-solving approach, you can enhance your understanding of quadratic equations and improve your ability to solve them effectively.

Conclusion

In conclusion, solving quadratic equations is a fundamental skill in mathematics with wide-ranging applications. By mastering techniques like factoring and the quadratic formula, and by understanding the graphical interpretation of these equations, you can confidently tackle a variety of problems. Remember to practice regularly and apply these concepts to real-world scenarios to solidify your understanding. With dedication and the right approach, quadratic equations become not just solvable, but also a powerful tool in your mathematical arsenal.