Solve The Equation: 21 X 2 − 2 X + 1 21 = 0 21x^2 - 2x + \frac{1}{21} = 0 21 X 2 − 2 X + 21 1 ​ = 0

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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 ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and xx is the variable.

In this article, we will focus on solving the quadratic equation 21x2−2x+121=021x^2 - 2x + \frac{1}{21} = 0. This equation is a classic example of a quadratic equation, and it can be solved using various methods such as factoring, completing the square, and the quadratic formula.

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=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. The quadratic formula is derived from the fact that a quadratic equation can be written in the form of (x−r)(x−s)=0(x - r)(x - s) = 0, where rr and ss are the roots of the equation.

To apply the quadratic formula, we need to identify the values of aa, bb, and cc in the given equation. In the equation 21x2−2x+121=021x^2 - 2x + \frac{1}{21} = 0, we have a=21a = 21, b=−2b = -2, and c=121c = \frac{1}{21}.

Applying the Quadratic Formula

Now that we have identified the values of aa, bb, and cc, we can apply the quadratic formula to solve the equation. Plugging in the values, we get:

x=−(−2)±(−2)2−4(21)(121)2(21)x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(21)(\frac{1}{21})}}{2(21)}

Simplifying the expression, we get:

x=2±4−442x = \frac{2 \pm \sqrt{4 - 4}}{42}

x=2±042x = \frac{2 \pm \sqrt{0}}{42}

x=242x = \frac{2}{42}

x=121x = \frac{1}{21}

Alternative Methods of Solving Quadratic Equations

While the quadratic formula is a powerful tool for solving quadratic equations, it is not the only method available. There are several alternative methods that can be used to solve quadratic equations, including factoring and completing the square.

Factoring

Factoring is a method of solving quadratic equations by expressing the equation as a product of two binomials. To factor a quadratic equation, we need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

In the equation 21x2−2x+121=021x^2 - 2x + \frac{1}{21} = 0, we can factor the equation as follows:

21x2−2x+121=(3x−17)(7x−13)=021x^2 - 2x + \frac{1}{21} = (3x - \frac{1}{7})(7x - \frac{1}{3}) = 0

Solving for xx, we get:

3x−17=03x - \frac{1}{7} = 0 or 7x−13=07x - \frac{1}{3} = 0

x=121x = \frac{1}{21} or x=121x = \frac{1}{21}

Completing the Square

Completing the square is a method of solving quadratic equations by expressing the equation in the form of (x−r)2=s(x - r)^2 = s. To complete the square, we need to add and subtract a constant term to the equation.

In the equation 21x2−2x+121=021x^2 - 2x + \frac{1}{21} = 0, we can complete the square as follows:

21x2−2x+121=21(x2−121x)+12121x^2 - 2x + \frac{1}{21} = 21(x^2 - \frac{1}{21}x) + \frac{1}{21}

21(x2−121x)+121=21(x2−121x+1441)−1441+12121(x^2 - \frac{1}{21}x) + \frac{1}{21} = 21(x^2 - \frac{1}{21}x + \frac{1}{441}) - \frac{1}{441} + \frac{1}{21}

21(x2−121x+1441)−1441+121=21(x−121)2−1441+12121(x^2 - \frac{1}{21}x + \frac{1}{441}) - \frac{1}{441} + \frac{1}{21} = 21(x - \frac{1}{21})^2 - \frac{1}{441} + \frac{1}{21}

21(x−121)2−1441+121=021(x - \frac{1}{21})^2 - \frac{1}{441} + \frac{1}{21} = 0

Solving for xx, we get:

21(x−121)2=1441−12121(x - \frac{1}{21})^2 = \frac{1}{441} - \frac{1}{21}

21(x−121)2=−12121(x - \frac{1}{21})^2 = -\frac{1}{21}

(x−121)2=−121(x - \frac{1}{21})^2 = -\frac{1}{21}

x−121=±−121x - \frac{1}{21} = \pm \sqrt{-\frac{1}{21}}

x=121±−121x = \frac{1}{21} \pm \sqrt{-\frac{1}{21}}

Conclusion

In this article, we have discussed the quadratic equation 21x2−2x+121=021x^2 - 2x + \frac{1}{21} = 0 and its solutions using various methods such as the quadratic formula, factoring, and completing the square. We have also discussed the importance of quadratic equations in various fields and the different methods available for solving them.

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the solutions to a quadratic equation in the form of x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

In addition to the quadratic formula, there are several alternative methods available for solving quadratic equations, including factoring and completing the square. Factoring is a method of solving quadratic equations by expressing the equation as a product of two binomials, while completing the square is a method of solving quadratic equations by expressing the equation in the form of (x−r)2=s(x - r)^2 = s.

We hope that this article has provided a comprehensive overview of the quadratic equation 21x2−2x+121=021x^2 - 2x + \frac{1}{21} = 0 and its solutions using various methods. We also hope that this article has provided a better understanding of the importance of quadratic equations in various fields and the different methods available for solving them.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In our previous article, we discussed the quadratic equation 21x2−2x+121=021x^2 - 2x + \frac{1}{21} = 0 and its solutions using various methods such as the quadratic formula, factoring, and completing the square.

In this article, we will provide a Q&A section to address some of the frequently asked questions about quadratic equations. We will cover topics such as the quadratic formula, factoring, completing the square, and more.

Q1: What is the quadratic formula?

A1: 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=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Q2: How do I apply the quadratic formula?

A2: To apply the quadratic formula, you need to identify the values of aa, bb, and cc in the given equation. Then, you can plug these values into the quadratic formula to find the solutions.

Q3: What is factoring?

A3: Factoring is a method of solving quadratic equations by expressing the equation as a product of two binomials. To factor a quadratic equation, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Q4: How do I factor a quadratic equation?

A4: To factor a quadratic equation, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term. Then, you can express the equation as a product of two binomials.

Q5: What is completing the square?

A5: Completing the square is a method of solving quadratic equations by expressing the equation in the form of (x−r)2=s(x - r)^2 = s. To complete the square, you need to add and subtract a constant term to the equation.

Q6: How do I complete the square?

A6: To complete the square, you need to add and subtract a constant term to the equation. Then, you can express the equation in the form of (x−r)2=s(x - r)^2 = s.

Q7: What is the difference between the quadratic formula and factoring?

A7: The quadratic formula and factoring are two different methods of solving quadratic equations. The quadratic formula provides a general solution to a quadratic equation, while factoring provides a specific solution.

Q8: Can I use the quadratic formula and factoring together?

A8: Yes, you can use the quadratic formula and factoring together to solve a quadratic equation. The quadratic formula can be used to find the solutions to a quadratic equation, while factoring can be used to express the equation as a product of two binomials.

Q9: What is the importance of quadratic equations in real-life applications?

A9: Quadratic equations are used in various fields such as physics, engineering, and economics. They are used to model real-world problems such as projectile motion, electrical circuits, and population growth.

Q10: Can I use quadratic equations to solve problems in other areas of mathematics?

A10: Yes, you can use quadratic equations to solve problems in other areas of mathematics such as algebra, geometry, and trigonometry. Quadratic equations are a fundamental concept in mathematics, and they can be used to solve a wide range of problems.

Conclusion

In this article, we have provided a Q&A section to address some of the frequently asked questions about quadratic equations. We have covered topics such as the quadratic formula, factoring, completing the square, and more. We hope that this article has provided a better understanding of quadratic equations and their applications in real-life problems.

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. The quadratic formula, factoring, and completing the square are three different methods of solving quadratic equations, and they can be used together to solve a wide range of problems.

We hope that this article has provided a comprehensive overview of quadratic equations and their applications in real-life problems. We also hope that this article has provided a better understanding of the importance of quadratic equations in mathematics and their role in solving real-world problems.