(a-1)x^2-(a+1)x+a+1 0

Jasonbradley

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Jun 16, 2024

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Solving the Quadratic Equation: (a-1)x² - (a+1)x + a + 1 = 0

This article explores the solution of the quadratic equation (a-1)x² - (a+1)x + a + 1 = 0, where 'a' is a constant. We will analyze the equation, find its roots, and discuss the different scenarios based on the value of 'a'.

Understanding the Equation

The given equation is a quadratic equation in the form of ax² + bx + c = 0, where:

• a = (a-1)
• b = -(a+1)
• c = a + 1

To solve this equation, we can utilize the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

Solving for the Roots

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

x = [(a+1) ± √((-a-1)² - 4(a-1)(a+1))] / 2(a-1)

Simplifying the expression under the square root:

x = [(a+1) ± √(a² + 2a + 1 - 4a² + 4)] / 2(a-1)

x = [(a+1) ± √(-3a² + 2a + 5)] / 2(a-1)

Analyzing the Solutions

The nature of the roots depends on the discriminant (the expression under the square root):

• Discriminant > 0: The equation has two distinct real roots.
• Discriminant = 0: The equation has one real root (a double root).
• Discriminant < 0: The equation has no real roots, but two complex roots.

Scenarios based on the value of 'a':

1. a = 1: The equation becomes undefined as the denominator becomes zero. This indicates that there is no solution for this specific value of 'a'.

2. a ≠ 1: The equation is defined, and we can analyze the discriminant:

   • Discriminant > 0: The equation has two distinct real roots. This occurs when -3a² + 2a + 5 > 0. Solving this inequality gives us a range of values for 'a' where this scenario holds true.
   • Discriminant = 0: The equation has one real root (a double root). This occurs when -3a² + 2a + 5 = 0. Solving this equation will give us the specific value of 'a' for which this scenario occurs.
   • Discriminant < 0: The equation has two complex roots. This occurs when -3a² + 2a + 5 < 0. Solving this inequality gives us a range of values for 'a' where this scenario holds true.

Conclusion

By applying the quadratic formula and analyzing the discriminant, we can determine the nature and values of the roots of the equation (a-1)x² - (a+1)x + a + 1 = 0 for different values of 'a'. It's important to consider the specific value of 'a' and analyze the discriminant to understand the solution behavior of the equation.