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

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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 specific quadratic equation using the quadratic formula. The equation we will be solving is 5x2−8x+5=05x^2 - 8x + 5 = 0. We will also discuss the importance of writing the solutions in a specific form, where rr, ss, and tt are integers, and the fractions are in their simplest form.

What is the Quadratic Formula?

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form ax2+bx+c=0ax^2 + bx + c = 0. The formula is given by:

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

where aa, bb, and cc are the coefficients of the quadratic equation.

Applying the Quadratic Formula

To solve the quadratic equation 5x2−8x+5=05x^2 - 8x + 5 = 0, we will use the quadratic formula. We have a=5a = 5, b=−8b = -8, and c=5c = 5. Plugging these values into the quadratic formula, we get:

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

Simplifying the expression, we get:

x=8±64−10010x = \frac{8 \pm \sqrt{64 - 100}}{10}

x=8±−3610x = \frac{8 \pm \sqrt{-36}}{10}

x=8±6i10x = \frac{8 \pm 6i}{10}

where ii is the imaginary unit, which is defined as the square root of −1-1.

Writing the Solutions in the Required Form

The solutions to the quadratic equation are given by:

x=8+6i10x = \frac{8 + 6i}{10}

x=8−6i10x = \frac{8 - 6i}{10}

We can simplify these expressions by dividing both the numerator and the denominator by their greatest common divisor, which is 22. This gives us:

x=4+3i5x = \frac{4 + 3i}{5}

x=4−3i5x = \frac{4 - 3i}{5}

These are the solutions to the quadratic equation 5x2−8x+5=05x^2 - 8x + 5 = 0.

Importance of Writing Solutions in the Required Form

Writing the solutions in the required form is important because it allows us to easily identify the real and imaginary parts of the solutions. In this case, the solutions are complex numbers, and writing them in the required form helps us to see that the real part is 4/54/5 and the imaginary part is ±3/5\pm 3/5.

Conclusion

Solving quadratic equations is an important skill in mathematics, and the quadratic formula is a powerful tool for solving these equations. In this article, we have solved the quadratic equation 5x2−8x+5=05x^2 - 8x + 5 = 0 using the quadratic formula and written the solutions in the required form. We have also discussed the importance of writing solutions in the required form, which allows us to easily identify the real and imaginary parts of the solutions.

Future Directions

In the future, we can use the quadratic formula to solve more complex quadratic equations. We can also explore other methods for solving quadratic equations, such as factoring and the quadratic formula. Additionally, we can use the quadratic formula to solve systems of quadratic equations and to find the roots of polynomial equations.

References

  • [1] "Quadratic Formula" by Math Open Reference. Retrieved February 26, 2024.
  • [2] "Solving Quadratic Equations" by Khan Academy. Retrieved February 26, 2024.

Glossary

  • Quadratic Equation: An equation of the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.
  • Quadratic Formula: A mathematical formula that provides the solutions to a quadratic equation.
  • Imaginary Unit: A complex number that is defined as the square root of −1-1.
  • Complex Number: A number that can be expressed in the form a+bia + bi, where aa and bb are real numbers and ii is the imaginary unit.

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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 comprehensive Q&A guide to help you understand quadratic equations and how to solve them.

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 (usually x) is two. It is typically written in the form ax^2 + bx + c = 0, where a, b, and c are constants.

Q: What is the Quadratic Formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

x = (-b ± √(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, plug these values into the formula and simplify the expression to find the solutions.

Q: What If the Quadratic Equation Has No Real Solutions?

A: If the quadratic equation has no real solutions, it means that the discriminant (b^2 - 4ac) is negative. In this case, the solutions will be complex numbers, which can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit.

Q: How Do I Write Complex Solutions in the Required Form?

A: To write complex solutions in the required form, you need to simplify the expression by dividing both the numerator and the denominator by their greatest common divisor. This will give you the solutions in the form a + bi, where a and b are real numbers and i is the imaginary unit.

Q: What is the Importance of Writing Solutions in the Required Form?

A: Writing solutions in the required form is important because it allows you to easily identify the real and imaginary parts of the solutions. This is particularly useful when working with complex numbers.

Q: Can I Use the Quadratic Formula to Solve Systems of Quadratic Equations?

A: Yes, you can use the quadratic formula to solve systems of quadratic equations. However, you need to be careful when solving systems of equations, as the solutions may not be unique.

Q: Can I Use the Quadratic Formula to Find the Roots of Polynomial Equations?

A: Yes, you can use the quadratic formula to find the roots of polynomial equations. However, you need to be careful when working with polynomial equations, as the solutions may not be unique.

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 identifying the coefficients a, b, and c correctly
  • Not simplifying the expression correctly
  • Not writing complex solutions in the required form
  • Not being careful when working with complex numbers

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we have provided a comprehensive Q&A guide to help you understand quadratic equations and how to solve them. We hope this guide has been helpful in answering your questions and providing you with a better understanding of quadratic equations.

Glossary

  • Quadratic Equation: A polynomial equation of degree two, which means the highest power of the variable (usually x) is two.
  • Quadratic Formula: A mathematical formula that provides the solutions to a quadratic equation.
  • Complex Number: A number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.
  • Imaginary Unit: A complex number that is defined as the square root of -1.