Solve The Equation: X 2 + ( A B C − C A B ) X − 1 = 0 X^2 + \left(\frac{ab}{c} - \frac{c}{ab}\right)x - 1 = 0 X 2 + ( C Ab ​ − Ab C ​ ) X − 1 = 0

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Introduction

In this article, we will delve into the world of quadratic equations and explore a specific equation of the form x2+(abc−cab)x−1=0x^2 + \left(\frac{ab}{c} - \frac{c}{ab}\right)x - 1 = 0. We will break down the equation, identify its components, and use various techniques to solve for the value of xx. This equation may seem daunting at first, but with a clear understanding of the concepts and a step-by-step approach, we can successfully solve it.

Understanding the Equation

The given equation is a quadratic equation in the form of ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants. In this case, the equation is:

x2+(abc−cab)x−1=0x^2 + \left(\frac{ab}{c} - \frac{c}{ab}\right)x - 1 = 0

To solve this equation, we need to identify the values of aa, bb, and cc. In this equation, a=1a = 1, b=(abc−cab)b = \left(\frac{ab}{c} - \frac{c}{ab}\right), and c=−1c = -1.

Simplifying the Equation

Before we can solve the equation, we need to simplify it by combining like terms. We can start by simplifying the expression for bb:

b=(abc−cab)b = \left(\frac{ab}{c} - \frac{c}{ab}\right)

To simplify this expression, we can multiply both the numerator and denominator by abab to get rid of the fraction:

b=a2b2−c2abcb = \frac{a^2b^2 - c^2}{abc}

Now, we can substitute this expression for bb back into the original equation:

x2+a2b2−c2abcx−1=0x^2 + \frac{a^2b^2 - c^2}{abc}x - 1 = 0

Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by:

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

In this case, we have a=1a = 1, b=a2b2−c2abcb = \frac{a^2b^2 - c^2}{abc}, and c=−1c = -1. We can substitute these values into the quadratic formula:

x=−a2b2−c2abc±(a2b2−c2abc)2−4(1)(−1)2(1)x = \frac{-\frac{a^2b^2 - c^2}{abc} \pm \sqrt{\left(\frac{a^2b^2 - c^2}{abc}\right)^2 - 4(1)(-1)}}{2(1)}

Simplifying the Quadratic Formula

To simplify the quadratic formula, we can start by simplifying the expression under the square root:

(a2b2−c2abc)2−4(1)(−1)\left(\frac{a^2b^2 - c^2}{abc}\right)^2 - 4(1)(-1)

We can expand the square and simplify the expression:

(a2b2−c2)2a2b2c2+4\frac{(a^2b^2 - c^2)^2}{a^2b^2c^2} + 4

Now, we substitute this expression back into the quadratic formula:

x=−a2b2−c2abc±(a2b2−c2)2a2b2c2+42x = \frac{-\frac{a^2b^2 - c^2}{abc} \pm \sqrt{\frac{(a^2b^2 - c^2)^2}{a^2b^2c^2} + 4}}{2}

Solving for x

To solve for xx, we can simplify the expression under the square root:

(a2b2−c2)2a2b2c2+4\frac{(a^2b^2 - c^2)^2}{a^2b^2c^2} + 4

We can expand the square and simplify the expression:

a4b4−2a2b2c2+c4a2b2c2+4\frac{a^4b^4 - 2a^2b^2c^2 + c^4}{a^2b^2c^2} + 4

Now, we can substitute this expression back into the quadratic formula:

x=−a2b2−c2abc±a4b4−2a2b2c2+c4a2b2c2+42x = \frac{-\frac{a^2b^2 - c^2}{abc} \pm \sqrt{\frac{a^4b^4 - 2a^2b^2c^2 + c^4}{a^2b^2c^2} + 4}}{2}

Conclusion

In this article, we have solved the quadratic equation x2+(abc−cab)x−1=0x^2 + \left(\frac{ab}{c} - \frac{c}{ab}\right)x - 1 = 0 using the quadratic formula. We have broken down the equation, identified its components, and used various techniques to simplify the expression and solve for the value of xx. This equation may seem daunting at first, but with a clear understanding of the concepts and a step-by-step approach, we can successfully solve it.

Final Answer

The final answer is:

x=−a2b2−c2abc±a4b4−2a2b2c2+c4a2b2c2+42x = \frac{-\frac{a^2b^2 - c^2}{abc} \pm \sqrt{\frac{a^4b^4 - 2a^2b^2c^2 + c^4}{a^2b^2c^2} + 4}}{2}

Note: The final answer is a complex expression that involves the variables aa, bb, and cc. The solution is valid for all values of aa, bb, and cc except when a=0a = 0, b=0b = 0, or c=0c = 0.

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Introduction

In our previous article, we solved the quadratic equation x2+(abc−cab)x−1=0x^2 + \left(\frac{ab}{c} - \frac{c}{ab}\right)x - 1 = 0 using the quadratic formula. However, we understand that some readers may still have questions about the equation and its solution. In this article, we will address some of the most frequently asked questions about quadratic equations and provide additional insights and explanations.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable (in this case, xx) 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.

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, we need to examine the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by:

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

Q: How do I apply the quadratic formula to a quadratic equation?

A: To apply the quadratic formula to a quadratic equation, we need to identify the values of aa, bb, and cc. We can then substitute these values into the quadratic formula and simplify the expression to find the solutions.

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. In other words, a quadratic equation has a squared term, while a linear equation does not.

Q: Can I use the quadratic formula to solve a cubic equation?

A: No, the quadratic formula is only applicable to quadratic equations. To solve a cubic equation, we need to use a different method, such as factoring or the cubic formula.

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula. It determines the nature of the solutions to the equation. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

Q: Can I use the quadratic formula to solve a quadratic equation with complex coefficients?

A: Yes, the quadratic formula can be used to solve a quadratic equation with complex coefficients. However, we need to be careful when simplifying the expression to avoid introducing extraneous solutions.

Conclusion

In this article, we have addressed some of the most frequently asked questions about quadratic equations and provided additional insights and explanations. We hope that this article has been helpful in clarifying any doubts or misconceptions about quadratic equations and their solutions.

Final Tips

  • Always check the discriminant to determine the nature of the solutions to a quadratic equation.
  • Use the quadratic formula to solve quadratic equations with real or complex coefficients.
  • Be careful when simplifying the expression to avoid introducing extraneous solutions.
  • Practice solving quadratic equations to become more comfortable with the quadratic formula and its applications.