Solve The Following Quadratic Function Using The Quadratic Formula:$y = 5m^2 + 3m - 7$

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Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. In this article, we will focus on solving quadratic equations using the quadratic formula.

What is the Quadratic Formula?

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where aa, bb, and cc are the coefficients of the quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0.

Understanding the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is based on the concept of completing the square, which involves rewriting the quadratic equation in a perfect square form. The formula provides two solutions for the quadratic equation, which are given by the plus and minus signs in the formula.

Solving the Quadratic Function

Now, let's solve the quadratic function y=5m2+3m−7y = 5m^2 + 3m - 7 using the quadratic formula. To do this, we need to rewrite the equation in the form am2+bm+c=0am^2 + bm + c = 0. We can do this by subtracting 7 from both sides of the equation:

5m2+3m−7=05m^2 + 3m - 7 = 0

Now, we can identify the coefficients aa, bb, and cc:

a=5,b=3,c=−7a = 5, b = 3, c = -7

Applying the Quadratic Formula

Now, we can apply the quadratic formula to solve the equation:

m=−b±b2−4ac2am = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Substituting the values of aa, bb, and cc, we get:

m=−3±32−4(5)(−7)2(5)m = \frac{-3 \pm \sqrt{3^2 - 4(5)(-7)}}{2(5)}

Simplifying the expression, we get:

m=−3±9+14010m = \frac{-3 \pm \sqrt{9 + 140}}{10}

m=−3±14910m = \frac{-3 \pm \sqrt{149}}{10}

Simplifying the Solutions

The quadratic formula provides two solutions for the equation, which are given by the plus and minus signs in the formula. We can simplify the solutions by evaluating the square root:

m=−3±12.20710m = \frac{-3 \pm 12.207}{10}

The two solutions are:

m=−3+12.20710=1.1207m = \frac{-3 + 12.207}{10} = 1.1207

m=−3−12.20710=−1.5207m = \frac{-3 - 12.207}{10} = -1.5207

Conclusion

In this article, we have solved the quadratic function y=5m2+3m−7y = 5m^2 + 3m - 7 using the quadratic formula. We have identified the coefficients aa, bb, and cc, and applied the quadratic formula to solve the equation. The quadratic formula provides two solutions for the equation, which are given by the plus and minus signs in the formula. We have simplified the solutions by evaluating the square root.

Real-World Applications

Quadratic equations have numerous real-world applications in various fields such as physics, engineering, and economics. Some examples of quadratic equations include:

  • Projectile motion: The trajectory of a projectile under the influence of gravity is a quadratic equation.
  • Optimization problems: Quadratic equations are used to optimize functions in various fields such as economics and engineering.
  • Signal processing: Quadratic equations are used in signal processing to filter out noise and extract useful information.

Common Mistakes

When solving quadratic equations using the quadratic formula, there are several common mistakes to avoid:

  • Incorrect identification of coefficients: Make sure to identify the coefficients aa, bb, and cc correctly.
  • Incorrect application of the formula: Make sure to apply the quadratic formula correctly, including the plus and minus signs.
  • Incorrect simplification of solutions: Make sure to simplify the solutions correctly, including the evaluation of the square root.

Conclusion

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In our previous article, we discussed how to solve quadratic equations using the quadratic formula. In this article, we will provide a Q&A guide to help you understand quadratic equations better.

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 is two. It is typically written in the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are the coefficients of the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to identify the coefficients aa, bb, and cc in the quadratic equation. Then, you can plug these values into the quadratic formula to get the solutions.

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

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

  • Incorrect identification of coefficients: Make sure to identify the coefficients aa, bb, and cc correctly.
  • Incorrect application of the formula: Make sure to apply the quadratic formula correctly, including the plus and minus signs.
  • Incorrect simplification of solutions: Make sure to simplify the solutions correctly, including the evaluation of the square root.

Q: Can I solve quadratic equations without using the quadratic formula?

A: Yes, you can solve quadratic equations without using the quadratic formula. Some methods include:

  • Factoring: If the quadratic equation can be factored into the product of two binomials, you can solve it by setting each factor equal to zero.
  • Completing the square: You can rewrite the quadratic equation in a perfect square form and then solve it by setting the square equal to zero.
  • Graphing: You can graph the quadratic equation and find the solutions by finding the x-intercepts of the graph.

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

A: Quadratic equations have numerous real-world applications in various fields such as physics, engineering, and economics. Some examples include:

  • Projectile motion: The trajectory of a projectile under the influence of gravity is a quadratic equation.
  • Optimization problems: Quadratic equations are used to optimize functions in various fields such as economics and engineering.
  • Signal processing: Quadratic equations are used in signal processing to filter out noise and extract useful information.

Q: Can I use a calculator to solve quadratic equations?

A: Yes, you can use a calculator to solve quadratic equations. Most calculators have a built-in quadratic formula function that you can use to solve the equation.

Q: What are the limitations of the quadratic formula?

A: The quadratic formula has some limitations. For example:

  • Complex solutions: The quadratic formula can produce complex solutions, which are not real numbers.
  • Non-real solutions: The quadratic formula can produce non-real solutions, which are not real numbers.
  • No solution: The quadratic formula can produce no solution, which means that the equation has no real solutions.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. The quadratic formula is a powerful tool for solving quadratic equations, but it has some limitations. We hope that this Q&A guide has helped you understand quadratic equations better and has provided you with a better understanding of the quadratic formula.