Make sure to identify the coefficients correctly

Oct 13, 2025 by miodeya.com 48 views

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 $$x^2 - 15x + 50 = 0$$ using the quadratic formula. The quadratic formula is a powerful tool that can be used to solve quadratic equations of the form $$ax^2 + bx + c = 0$$.

The Quadratic Formula

The quadratic formula is given by:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula is derived from the process of completing the square, which involves manipulating the quadratic equation to express it in the form of a perfect square trinomial.

Step 1: Identify the Coefficients

To use the quadratic formula, we need to identify the coefficients of the quadratic equation. In the equation $$x^2 - 15x + 50 = 0$$, we have:

$a = 1$
$b = -15$
$c = 50$

Step 2: Plug in the Coefficients

Now that we have identified the coefficients, we can plug them into the quadratic formula:

x = \frac{-(-15) \pm \sqrt{(-15)^2 - 4(1)(50)}}{2(1)}

Step 3: Simplify the Expression

Simplifying the expression inside the square root, we get:

x = \frac{15 \pm \sqrt{225 - 200}}{2}

x = \frac{15 \pm \sqrt{25}}{2}

x = \frac{15 \pm 5}{2}

Step 4: Solve for x

Now that we have simplified the expression, we can solve for x:

x = \frac{15 + 5}{2}

x = \frac{20}{2}

x = 10

x = \frac{15 - 5}{2}

x = \frac{10}{2}

x = 5

Conclusion

In this article, we have used the quadratic formula to solve the quadratic equation $$x^2 - 15x + 50 = 0$$. We have identified the coefficients, plugged them into the quadratic formula, simplified the expression, and solved for x. The solutions to the equation are x = 10 and x = 5.

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 cost-benefit analysis.

Tips and Tricks

Here are some tips and tricks for solving quadratic equations using the quadratic formula:

Make sure to identify the coefficients correctly: The coefficients of the quadratic equation are crucial in determining the solutions.
Simplify the expression inside the square root: Simplifying the expression inside the square root can make it easier to solve for x.
Check for extraneous solutions: Quadratic equations can have extraneous solutions, which are solutions that do not satisfy the original equation.

Common Mistakes

Here are some common mistakes to avoid when solving quadratic equations using the quadratic formula:

Incorrectly identifying the coefficients: Incorrectly identifying the coefficients can lead to incorrect solutions.
Not simplifying the expression inside the square root: Not simplifying the expression inside the square root can make it difficult to solve for x.
Not checking for extraneous solutions: Not checking for extraneous solutions can lead to incorrect solutions.

Conclusion

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 provide a Q&A guide to help you understand quadratic equations and how to solve them using the quadratic formula.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable (usually x) is two. It is typically written in the form of $$ax^2 + bx + c = 0$$, where a, b, and c are constants.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool that can be used to solve quadratic equations of the form $$ax^2 + bx + c = 0$$. It is given by:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula, you need to identify the coefficients of the quadratic equation, which are a, b, and c. Then, you plug these values into the quadratic formula and simplify the expression to find the solutions.

Q: What are the steps to solve a quadratic equation using the quadratic formula?

A: The steps to solve a quadratic equation using the quadratic formula are:

Identify the coefficients of the quadratic equation (a, b, and c).
Plug these values into the quadratic formula.
Simplify the expression inside the square root.
Solve for x.

Q: What are the different types of solutions to a quadratic equation?

A: The solutions to a quadratic equation can be:

Real and distinct: These are two distinct real numbers that satisfy the equation.
Real and equal: These are two equal real numbers that satisfy the equation.
Complex: These are complex numbers that satisfy the equation.

Q: How do I determine the type of solution to a quadratic equation?

A: To determine the type of solution to a quadratic equation, you need to look at the discriminant, which is the expression inside the square root in the quadratic formula. If the discriminant is:

Positive: The solutions are real and distinct.
Zero: The solutions are real and equal.
Negative: The solutions are complex.

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

A: Some common mistakes to avoid when solving quadratic equations using the quadratic formula are:

Incorrectly identifying the coefficients: Incorrectly identifying the coefficients can lead to incorrect solutions.
Not simplifying the expression inside the square root: Not simplifying the expression inside the square root can make it difficult to solve for x.
Not checking for extraneous solutions: Not checking for extraneous solutions can lead to incorrect solutions.

Q: How can I practice solving quadratic equations using the quadratic formula?

A: You can practice solving quadratic equations using the quadratic formula by:

Solving sample problems: Try solving sample problems to get a feel for how to use the quadratic formula.
Using online resources: There are many online resources available that can help you practice solving quadratic equations using the quadratic formula.
Working with a tutor or teacher: Working with a tutor or teacher can help you get personalized feedback and guidance.

Conclusion

In conclusion, solving quadratic equations using the quadratic formula is a powerful tool that can be used to solve quadratic equations of the form $$ax^2 + bx + c = 0$$. By understanding the quadratic formula and how to use it, you can solve quadratic equations with ease. Remember to identify the coefficients, plug them into the quadratic formula, simplify the expression, and solve for x. With practice and patience, solving quadratic equations using the quadratic formula can become a breeze.