(x-3)%3D(x%2B5)(x-1)-solution.jpeg "(2x-1)(x-3)=(x+5)(x-1) Solution")
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).
Expanding the Equation
First, we need to expand both sides of the equation by multiplying out the brackets:
- Left side: (2x-1)(x-3) = 2x² - 6x - x + 3 = 2x² - 7x + 3
- Right side: (x+5)(x-1) = x² - x + 5x - 5 = x² + 4x - 5
Now our equation looks like this: 2x² - 7x + 3 = x² + 4x - 5
Rearranging the Equation
To solve for x, we need to rearrange the equation so that all the terms are on one side and the equation equals zero.
Let's subtract x² + 4x - 5 from both sides:
- 2x² - 7x + 3 - (x² + 4x - 5) = 0
Simplifying, we get:
- x² - 11x + 8 = 0
Solving the Quadratic Equation
Now we have a quadratic equation in the form ax² + bx + c = 0. There are a couple of ways to solve this:
- Factoring: Try to find two numbers that add up to -11 and multiply to 8. In this case, factoring is not straightforward.
- Quadratic Formula: The quadratic formula is a reliable way to solve any quadratic equation. It states:
x = (-b ± √(b² - 4ac)) / 2a
In our equation, a = 1, b = -11, and c = 8. Plugging 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 are:
- x = (11 + √89) / 2
- x = (11 - √89) / 2
Conclusion
We have successfully solved the equation (2x-1)(x-3) = (x+5)(x-1) by expanding the equation, rearranging it into a quadratic equation, and then solving the quadratic equation using the quadratic formula. The solutions are x = (11 + √89) / 2 and x = (11 - √89) / 2.