Type The Correct Answer In The Box.Solve The Following Quadratic Equation Using The Quadratic Formula: 5 X 2 − 8 X + 5 = 0 5x^2 - 8x + 5 = 0 5 X 2 − 8 X + 5 = 0 Write The Solutions In The Following Form, Where { R, S $}$, And { T $}$ Are Integers, And The

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields, including 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. In this article, we will focus on solving quadratic equations using 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 provides two solutions for the equation, which are given by the plus and minus signs in the formula.

Step-by-Step Solution

To solve the quadratic equation $5x^2 - 8x + 5 = 0$, we will use the quadratic formula. The first step is to identify the coefficients of the equation, which are $a = 5$, $b = -8$, and $c = 5$. Next, we will plug these values into the quadratic formula.

Plugging in the Values

Substituting the values of $a$, $b$, and $c$ into the quadratic formula, we get:

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

Simplifying the expression, we get:

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

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

Simplifying the Square Root

The square root of a negative number is an imaginary number, which is denoted by $i$. Therefore, we can simplify the expression as follows:

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

Simplifying the Expression

We can simplify the expression further by dividing both the real and imaginary parts by 10:

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

Conclusion

In this article, we have solved the quadratic equation $5x^2 - 8x + 5 = 0$ using the quadratic formula. The solutions are given by the expression $x = \frac{4}{5} \pm \frac{3}{5}i$. We have also discussed the importance of the quadratic formula in solving quadratic equations.

Real-World Applications

Quadratic equations have numerous applications in various fields, including physics, engineering, and economics. For example, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic equation. Similarly, the motion of a pendulum can be described using a quadratic equation.

Tips and Tricks

When solving quadratic equations using the quadratic formula, it is essential to remember the following tips and tricks:

• Make sure to identify the coefficients of the equation correctly.
• Plug the values of $a$, $b$, and $c$ into the quadratic formula.
• Simplify the expression carefully, especially when dealing with square roots.
• Be careful when dividing complex numbers.

Common Mistakes

When solving quadratic equations using the quadratic formula, it is easy to make mistakes. Some common mistakes include:

• Forgetting to identify the coefficients of the equation.
• Plugging in the wrong values into the quadratic formula.
• Simplifying the expression incorrectly.
• Forgetting to consider the imaginary part of the solution.

Conclusion

Frequently Asked Questions

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. 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.

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 to solve a quadratic equation?

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

Q: What are the solutions to a quadratic equation?

A: The solutions to a quadratic equation are given by the quadratic formula. The solutions are in the form of $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions. The quadratic formula provides two solutions for the equation, which are given by the plus and minus signs in the formula.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. The general form of a linear equation is $ax + b = 0$, where $a$ and $b$ are constants, and $x$ is the variable.

Q: Can a quadratic equation be solved using factoring?

A: Yes, a quadratic equation can be solved using factoring. However, factoring is not always possible, and the quadratic formula is a more general method for solving quadratic equations.

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

A: Quadratic equations have numerous applications in various fields, including physics, engineering, and economics. For example, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic equation. Similarly, the motion of a pendulum can be described using a quadratic equation.

Q: How do I simplify a quadratic equation?

A: To simplify a quadratic equation, you need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate any expressions inside parentheses.
2. Exponents: Evaluate any exponential expressions.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Q: What is the discriminant of a quadratic equation?

A: The discriminant of a quadratic equation is the expression $b^2 - 4ac$ under the square root in the quadratic formula. The discriminant determines the nature of the solutions to the equation.

Q: Can a quadratic equation have a negative discriminant?

A: Yes, a quadratic equation can have a negative discriminant. In this case, the equation has no real solutions, but it has complex solutions.

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

A: To determine the nature of the solutions to a quadratic equation, you need to examine the discriminant. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions, but it has complex solutions.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. The quadratic formula is a powerful tool for solving quadratic equations, and it provides two solutions for the equation. By following the steps outlined in this article, you can solve quadratic equations with ease. Remember to identify the coefficients of the equation correctly, plug in the values into the quadratic formula, simplify the expression carefully, and be careful when dividing complex numbers.