(x-3)%3D(x%2B5)(x-1).jpeg "(2x-1)(x-3)=(x+5)(x-1)")
Solving the Equation: (2x-1)(x-3) = (x+5)(x-1)
This article will guide you through the steps of solving the equation (2x-1)(x-3) = (x+5)(x-1).
Step 1: Expand both sides of the equation
We begin by expanding both sides of the equation using the distributive property (also known as FOIL method):
- Left side: (2x-1)(x-3) = 2x(x-3) - 1(x-3) = 2x² - 6x - x + 3 = 2x² - 7x + 3
- Right side: (x+5)(x-1) = x(x-1) + 5(x-1) = x² - x + 5x - 5 = x² + 4x - 5
Now, our equation becomes: 2x² - 7x + 3 = x² + 4x - 5
Step 2: Simplify the equation
To simplify, we will move all the terms to one side of the equation. Subtracting x² and 4x from both sides, and adding 5 to both sides, we get:
2x² - 7x + 3 - x² - 4x + 5 = 0
Simplifying further: x² - 11x + 8 = 0
Step 3: Solve the quadratic equation
We now have a quadratic equation in the form of ax² + bx + c = 0. To solve this, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
In our equation, a = 1, b = -11, and c = 8. Substituting these values into the quadratic formula:
x = (11 ± √((-11)² - 4 * 1 * 8)) / 2 * 1
x = (11 ± √(89)) / 2
Therefore, the solutions to the equation (2x-1)(x-3) = (x+5)(x-1) are:
- x = (11 + √89) / 2
- x = (11 - √89) / 2
Conclusion
By expanding the equation, simplifying, and using the quadratic formula, we have successfully solved the equation (2x-1)(x-3) = (x+5)(x-1) and obtained two distinct solutions.