Solve For The Roots In Simplest Form Using The Quadratic Formula:$\[ 2x^2 + 26 = 20x \\]Solve For \[$ X \$\]:$\[ X = \boxed{\text{(enter Your Answer Here)}} \\]

Feb 27, 2025 by ADMIN 161 views

**Solving Quadratic Equations: A Step-by-Step Guide to Finding Roots**

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 a quadratic equation using the quadratic formula, which is a powerful tool for finding the roots of a quadratic equation. We will use the equation $2x^2 + 26 = 20x$ as an example and walk through the steps to find the roots in simplest form.

Understanding Quadratic Equations

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. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.

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}

where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation. The quadratic formula is a general solution that works for all quadratic equations, regardless of whether they can be factored or not.

Solving the Quadratic Equation

Now, let's apply the quadratic formula to the equation $2x^2 + 26 = 20x$ . First, we need to rewrite the equation in the standard form $ax^2 + bx + c = 0$ . We can do this by subtracting $20x$ from both sides of the equation:

2x^2 - 20x + 26 = 0

Now, we can identify the coefficients $a$ , $b$ , and $c$ . In this case, $a = 2$ , $b = -20$ , and $c = 26$ .

Plugging in the Values

Next, we plug the values of $a$ , $b$ , and $c$ into the quadratic formula:

x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4(2)(26)}}{2(2)}

Simplifying the expression, we get:

x = \frac{20 \pm \sqrt{400 - 208}}{4}

x = \frac{20 \pm \sqrt{192}}{4}

Simplifying the Square Root

The square root of 192 can be simplified as follows:

\sqrt{192} = \sqrt{64 \times 3}

\sqrt{192} = 8\sqrt{3}

Now, we can substitute this value back into the expression for $x$ :

x = \frac{20 \pm 8\sqrt{3}}{4}

Simplifying the Expression

Finally, we can simplify the expression by dividing both the numerator and the denominator by 4:

x = 5 \pm 2\sqrt{3}

Conclusion

In this article, we used the quadratic formula to solve the quadratic equation $2x^2 + 26 = 20x$ . We walked through the steps to find the roots in simplest form, including plugging in the values of $a$ , $b$ , and $c$ into the quadratic formula and simplifying the expression. The final solution is $x = 5 \pm 2\sqrt{3}$ .

Real-World Applications

Quadratic equations have many 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 investment portfolios.

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 a general solution that works for all quadratic equations, regardless of whether they can be factored or not.
Simplify the expression: Simplifying the expression is an important step in solving quadratic equations. It can help to make the solution more manageable and easier to understand.
Check the solution: It is always a good idea to check the solution to a quadratic equation to make sure it is correct.

Common Mistakes

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

Not simplifying the expression: Failing to simplify the expression can make the solution more difficult to understand and may lead to errors.
Not checking the solution: Failing to check the solution can lead to errors and incorrect answers.
Not using the quadratic formula: Failing to use the quadratic formula can make it more difficult to solve the equation and may lead to errors.

Conclusion

In conclusion, solving quadratic equations is an important skill that has many real-world applications. The quadratic formula is a powerful tool for solving quadratic equations, and simplifying the expression is an important step in finding the solution. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving quadratic equations and apply this skill to a wide range of problems.

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 answer some of the most frequently asked questions about quadratic equations, including how to solve them, what the quadratic formula is, and how to apply it to real-world problems.

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.

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}

where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

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

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula and factoring are two different methods for solving quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while the quadratic formula involves using a formula to find the roots of the equation.

Q: When should I use the quadratic formula?

A: You should use the quadratic formula when the quadratic equation cannot be factored easily or when you need to find the roots of the equation quickly.

Q: Can I use the quadratic formula to solve quadratic equations with complex roots?

A: Yes, you can use the quadratic formula to solve quadratic equations with complex roots. The quadratic formula will give you the complex roots of the equation.

Q: How do I apply the quadratic formula to real-world problems?

A: The quadratic formula has many real-world applications, including physics, engineering, and economics. You can use the quadratic formula to model the motion of objects under the influence of gravity, friction, and other forces, design and optimize systems, and model economic systems.

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

A: Some common mistakes to avoid when using the quadratic formula include:

Not simplifying the expression
Not checking the solution
Not using the quadratic formula when it is the most efficient method

Q: Can I use the quadratic formula to solve quadratic equations with rational roots?

A: Yes, you can use the quadratic formula to solve quadratic equations with rational roots. The quadratic formula will give you the rational roots of the equation.

Q: How do I check the solution to a quadratic equation?

A: To check the solution to a quadratic equation, you need to plug the solution back into the original equation and simplify the expression. If the expression equals zero, then the solution is correct.

Q: Can I use the quadratic formula to solve quadratic equations with irrational roots?

A: Yes, you can use the quadratic formula to solve quadratic equations with irrational roots. The quadratic formula will give you the irrational roots of the equation.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations, and it has many real-world applications. By understanding how to use the quadratic formula and avoiding common mistakes, you can become proficient in solving quadratic equations and apply this skill to a wide range of problems.

Additional Resources

If you want to learn more about quadratic equations and the quadratic formula, here are some additional resources:

Textbooks: There are many textbooks available that cover quadratic equations and the quadratic formula in detail.
Online resources: There are many online resources available that provide tutorials, examples, and practice problems for quadratic equations and the quadratic formula.
Video tutorials: There are many video tutorials available that provide step-by-step instructions for solving quadratic equations and using the quadratic formula.

Practice Problems

Here are some practice problems to help you practice using the quadratic formula:

Problem 1: Solve the quadratic equation $x^2 + 5x + 6 = 0$ using the quadratic formula.
Problem 2: Solve the quadratic equation $x^2 - 3x - 4 = 0$ using the quadratic formula.
Problem 3: Solve the quadratic equation $x^2 + 2x - 3 = 0$ using the quadratic formula.