(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 Solve By Quadratic Formula

2 min read Jun 17, 2024
(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 Solve By Quadratic Formula

Solving the Equation (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 using the Quadratic Formula

This article will guide you through solving the equation (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 using the quadratic formula.

Expanding the Equation

First, we need to expand the equation to obtain a standard quadratic form:

(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0

Expanding each product:

  • x² - ax - bx + ab
  • x² - bx - cx + bc
  • x² - cx - ax + ac

Combining like terms:

3x² - 2(a+b+c)x + (ab+bc+ac) = 0

Applying the Quadratic Formula

Now, we have a quadratic equation in the form ax² + bx + c = 0. We can apply the quadratic formula to solve for x:

x = (-b ± √(b² - 4ac)) / 2a

In our equation:

  • a = 3
  • b = -2(a+b+c)
  • c = ab+bc+ac

Substitute these values into the quadratic formula:

x = (2(a+b+c) ± √((-2(a+b+c))² - 4 * 3 * (ab+bc+ac))) / (2 * 3)

Simplifying the Solution

After simplification, we arrive at the solution:

x = (a+b+c ± √((a+b+c)² - 3(ab+bc+ac))) / 3

Conclusion

By expanding the equation and applying the quadratic formula, we have successfully solved the equation (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 for x. The solution provides us with the two possible values of x that satisfy the given equation.