Using The Quadratic Formula, Solve $0=2x^2-8x+5$. Give Each Of Your Answers To 2 Decimal Places.

by ADMIN 99 views

Introduction

Quadratic equations are a fundamental concept in mathematics, and the quadratic formula is a powerful tool for solving them. In this article, we will explore how to use the quadratic formula to solve a quadratic equation, with a focus on the equation $0=2x^2-8x+5$. We will break down the steps involved in using the quadratic formula and provide a clear, step-by-step guide to solving the equation.

What is the Quadratic Formula?

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form $ax^2+bx+c=0$. The formula is given by:

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

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

How to Use the Quadratic Formula

To use the quadratic formula, we need to identify the values of $a$, $b$, and $c$ in the quadratic equation. In the equation $0=2x^2-8x+5$, we can see that $a=2$, $b=-8$, and $c=5$. Now, we can plug these values into the quadratic formula:

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

Simplifying the Quadratic Formula

To simplify the quadratic formula, we need to evaluate the expressions inside the square root and the fraction. Let's start with the expression inside the square root:

(−8)2−4(2)(5)(-8)^2-4(2)(5)

=64−40=64-40

=24=24

Now, we can plug this value back into the quadratic formula:

x=8±244x=\frac{8\pm\sqrt{24}}{4}

Simplifying the Square Root

To simplify the square root, we need to find the square root of $24$. The square root of $24$ is $\sqrt{24}=\sqrt{4\cdot6}=\sqrt{4}\cdot\sqrt{6}=2\sqrt{6}$. Now, we can plug this value back into the quadratic formula:

x=8±264x=\frac{8\pm2\sqrt{6}}{4}

Simplifying the Fraction

To simplify the fraction, we need to divide the numerator and denominator by their greatest common divisor. In this case, the greatest common divisor of $8$ and $4$ is $4$. So, we can divide both the numerator and denominator by $4$:

x=84±264x=\frac{8}{4}\pm\frac{2\sqrt{6}}{4}

=2±62=2\pm\frac{\sqrt{6}}{2}

Rounding the Solutions to 2 Decimal Places

To round the solutions to 2 decimal places, we need to evaluate the expressions $2+\frac{\sqrt{6}}{2}$ and $2-\frac{\sqrt{6}}{2}$. Let's start with the first expression:

2+622+\frac{\sqrt{6}}{2}

=2+0.82=2+0.82

=2.82=2.82

Now, let's evaluate the second expression:

2−622-\frac{\sqrt{6}}{2}

=2−0.82=2-0.82

=1.18=1.18

Conclusion

In this article, we have used the quadratic formula to solve the quadratic equation $0=2x^2-8x+5$. We have broken down the steps involved in using the quadratic formula and provided a clear, step-by-step guide to solving the equation. We have also rounded the solutions to 2 decimal places. The solutions to the equation are $x=2.82$ and $x=1.18$.

Real-World Applications of the Quadratic Formula

The quadratic formula has many real-world applications, including:

  • Physics: The quadratic formula is used to solve problems involving motion, such as the trajectory of a projectile.
  • Engineering: The quadratic formula is used to solve problems involving stress and strain on materials.
  • Computer Science: The quadratic formula is used to solve problems involving algorithms and data structures.

Common Mistakes to Avoid

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

  • Incorrectly identifying the values of a, b, and c: Make sure to identify the values of a, b, and c correctly in the quadratic equation.
  • Incorrectly simplifying the quadratic formula: Make sure to simplify the quadratic formula correctly, including evaluating the expressions inside the square root and the fraction.
  • Incorrectly rounding the solutions: Make sure to round the solutions to the correct number of decimal places.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. By following the steps outlined in this article, you can use the quadratic formula to solve a wide range of quadratic equations. Remember to identify the values of a, b, and c correctly, simplify the quadratic formula correctly, and round the solutions to the correct number of decimal places. With practice and patience, you can become proficient in using the quadratic formula to solve quadratic equations.

Introduction

The quadratic formula is a powerful tool for solving quadratic equations, but it can be intimidating for those who are new to it. In this article, we will answer some of the most frequently asked questions about the quadratic formula, including how to use it, common mistakes to avoid, and real-world applications.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form $ax^2+bx+c=0$. The formula is given by:

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

Q: How do I use the quadratic formula?

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

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

A: There are several common mistakes to avoid when using the quadratic formula, including:

  • Incorrectly identifying the values of a, b, and c: Make sure to identify the values of a, b, and c correctly in the quadratic equation.
  • Incorrectly simplifying the quadratic formula: Make sure to simplify the quadratic formula correctly, including evaluating the expressions inside the square root and the fraction.
  • Incorrectly rounding the solutions: Make sure to round the solutions to the correct number of decimal places.

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

A: The quadratic formula has many real-world applications, including:

  • Physics: The quadratic formula is used to solve problems involving motion, such as the trajectory of a projectile.
  • Engineering: The quadratic formula is used to solve problems involving stress and strain on materials.
  • Computer Science: The quadratic formula is used to solve problems involving algorithms and data structures.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with complex solutions. However, you need to be careful when simplifying the expression to find the solutions.

Q: How do I know if a quadratic equation has real or complex solutions?

A: To determine if a quadratic equation has real or complex solutions, you need to evaluate the expression inside the square root. If the expression is positive, then the equation has real solutions. If the expression is negative, then the equation has complex solutions.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with rational solutions. However, you need to be careful when simplifying the expression to find the solutions.

Q: How do I know if a quadratic equation has rational or irrational solutions?

A: To determine if a quadratic equation has rational or irrational solutions, you need to evaluate the expression inside the square root. If the expression is a perfect square, then the equation has rational solutions. If the expression is not a perfect square, then the equation has irrational solutions.

Q: Can I use the quadratic formula to solve quadratic equations with multiple solutions?

A: Yes, you can use the quadratic formula to solve quadratic equations with multiple solutions. However, you need to be careful when simplifying the expression to find the solutions.

Q: How do I know if a quadratic equation has multiple solutions?

A: To determine if a quadratic equation has multiple solutions, you need to evaluate the expression inside the square root. If the expression is equal to zero, then the equation has multiple solutions.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations, but it can be intimidating for those who are new to it. By following the steps outlined in this article, you can use the quadratic formula to solve a wide range of quadratic equations. Remember to identify the values of a, b, and c correctly, simplify the quadratic formula correctly, and round the solutions to the correct number of decimal places. With practice and patience, you can become proficient in using the quadratic formula to solve quadratic equations.