Solve The Following Quadratic Equation Using The Quadratic Formula:${ 5x^2 - 8x + 5 = 0 }$Write The Solutions In The Form Where { R, S, $}$ And { T $}$ Are Integers, And The Fractions Are In Simplest Form:$[ X =

Introduction to Quadratic Equations

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. The general form of a quadratic equation is ${ax^2 + bx + c = 0}$, where ${a}$, ${b}$, and ${c}$ are constants, and ${x}$ is the variable.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is a formula that provides the solutions to a quadratic equation in the form of ${x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$. The quadratic formula is derived from the process of completing the square, which involves manipulating the quadratic equation to express it in a perfect square form.

Step-by-Step Guide to Using the Quadratic Formula

To solve a quadratic equation using the quadratic formula, follow these steps:

1. Identify the coefficients: Identify the values of ${a}$, ${b}$, and ${c}$ in the quadratic equation.
2. Plug in the values: Plug in the values of ${a}$, ${b}$, and ${c}$ into the quadratic formula.
3. Simplify the expression: Simplify the expression under the square root, if possible.
4. Calculate the solutions: Calculate the two solutions using the quadratic formula.
5. Simplify the solutions: Simplify the solutions to express them in the form of integers or fractions.

Solving the Quadratic Equation

Now, let's solve the quadratic equation ${5x^2 - 8x + 5 = 0}$ using the quadratic formula.

Step 1: Identify the coefficients

The coefficients of the quadratic equation are ${a = 5}$, ${b = -8}$, and ${c = 5}$.

Step 2: Plug in the values

Plug in the values of ${a}$, ${b}$, and ${c}$ into the quadratic formula:

${x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(5)(5)}}{2(5)}}$

Step 3: Simplify the expression

Simplify the expression under the square root:

${x = \frac{8 \pm \sqrt{64 - 100}}{10}}$

${x = \frac{8 \pm \sqrt{-36}}{10}}$

Step 4: Calculate the solutions

Calculate the two solutions using the quadratic formula:

${x = \frac{8 \pm 6i}{10}}$

${x = \frac{4 \pm 3i}{5}}$

Step 5: Simplify the solutions

Simplify the solutions to express them in the form of integers or fractions:

${x = \frac{4}{5} \pm \frac{3}{5}i}$

Conclusion

In this article, we have discussed the quadratic formula and provided a step-by-step guide to using it to solve quadratic equations. We have also solved the quadratic equation ${5x^2 - 8x + 5 = 0}$ using the quadratic formula. The solutions to the equation are ${x = \frac{4}{5} \pm \frac{3}{5}i}$. The quadratic formula is a powerful tool for solving quadratic equations, and it is widely used in various fields of mathematics and science.

Real-World Applications of the Quadratic Formula

The quadratic formula has numerous real-world applications in various fields such as physics, engineering, and economics. Some of the applications of the quadratic formula include:

• Projectile motion: The quadratic formula is used to calculate the trajectory of a projectile under the influence of gravity.
• Optimization: The quadratic formula is used to optimize functions and find the maximum or minimum value of a function.
• Signal processing: The quadratic formula is used to filter signals and remove noise from a signal.
• Economics: The quadratic formula is used to model economic systems and predict economic trends.

Common Mistakes to Avoid

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

• Incorrectly identifying the coefficients: Make sure to identify the correct values of ${a}$, ${b}$, and ${c}$ in the quadratic equation.
• Simplifying the expression incorrectly: Make sure to simplify the expression under the square root correctly.
• Calculating the solutions incorrectly: Make sure to calculate the solutions correctly using the quadratic formula.
• Simplifying the solutions incorrectly: Make sure to simplify the solutions correctly to express them in the form of integers or fractions.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. It is widely used in various fields of mathematics and science, and it has numerous real-world applications. By following the step-by-step guide to using the quadratic formula, you can solve quadratic equations with ease and accuracy. Remember to avoid common mistakes and simplify the solutions correctly to express them in the form of integers or fractions.

Introduction

The quadratic formula is a powerful tool for solving quadratic equations, and it is widely used in various fields of mathematics and science. However, many students and professionals may have questions and doubts about using the quadratic formula. In this article, we will answer some of the most frequently asked questions about the quadratic formula.

Q1: What is the quadratic formula?

A1: The quadratic formula is a formula that provides the solutions to a quadratic equation in the form of ${x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$. It is derived from the process of completing the square, which involves manipulating the quadratic equation to express it in a perfect square form.

Q2: How do I use the quadratic formula?

A2: To use the quadratic formula, follow these steps:

1. Identify the coefficients: Identify the values of ${a}$, ${b}$, and ${c}$ in the quadratic equation.
2. Plug in the values: Plug in the values of ${a}$, ${b}$, and ${c}$ into the quadratic formula.
3. Simplify the expression: Simplify the expression under the square root, if possible.
4. Calculate the solutions: Calculate the two solutions using the quadratic formula.
5. Simplify the solutions: Simplify the solutions to express them in the form of integers or fractions.

Q3: What are the common mistakes to avoid when using the quadratic formula?

A3: Some common mistakes to avoid when using the quadratic formula include:

• Incorrectly identifying the coefficients: Make sure to identify the correct values of ${a}$, ${b}$, and ${c}$ in the quadratic equation.
• Simplifying the expression incorrectly: Make sure to simplify the expression under the square root correctly.
• Calculating the solutions incorrectly: Make sure to calculate the solutions correctly using the quadratic formula.
• Simplifying the solutions incorrectly: Make sure to simplify the solutions correctly to express them in the form of integers or fractions.

Q4: Can I use the quadratic formula to solve quadratic equations with complex coefficients?

A4: Yes, you can use the quadratic formula to solve quadratic equations with complex coefficients. However, you will need to use complex numbers and follow the rules of complex arithmetic.

Q5: How do I determine the nature of the solutions to a quadratic equation?

A5: To determine the nature of the solutions to a quadratic equation, you can use the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the solutions are real and distinct. If the discriminant is zero, the solutions are real and equal. If the discriminant is negative, the solutions are complex.

Q6: Can I use the quadratic formula to solve quadratic equations with rational coefficients?

A6: Yes, you can use the quadratic formula to solve quadratic equations with rational coefficients. However, you will need to simplify the solutions to express them in the form of integers or fractions.

Q7: How do I apply the quadratic formula in real-world problems?

A7: The quadratic formula has numerous real-world applications in various fields such as physics, engineering, and economics. Some examples of real-world problems that can be solved using the quadratic formula include:

• Projectile motion: The quadratic formula is used to calculate the trajectory of a projectile under the influence of gravity.
• Optimization: The quadratic formula is used to optimize functions and find the maximum or minimum value of a function.
• Signal processing: The quadratic formula is used to filter signals and remove noise from a signal.
• Economics: The quadratic formula is used to model economic systems and predict economic trends.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations, and it has numerous real-world applications. By following the step-by-step guide to using the quadratic formula and avoiding common mistakes, you can solve quadratic equations with ease and accuracy. Remember to simplify the solutions correctly to express them in the form of integers or fractions.