(x-3)%3Dx(4-x)-9.jpeg "(2x+1)(x-3)=x(4-x)-9")
Solving the Equation: (2x+1)(x-3) = x(4-x) - 9
This article will guide you through the steps involved in solving the equation (2x+1)(x-3) = x(4-x) - 9.
Step 1: Expand Both Sides
First, we need to expand both sides of the equation to simplify it:
- Left Side: (2x+1)(x-3) = 2x² - 6x + x - 3 = 2x² - 5x - 3
- Right Side: x(4-x) - 9 = 4x - x² - 9 = -x² + 4x - 9
Now, the equation becomes: 2x² - 5x - 3 = -x² + 4x - 9
Step 2: Combine Like Terms
To make the equation easier to solve, let's move all the terms to one side:
- Add x² to both sides: 3x² - 5x - 3 = 4x - 9
- Subtract 4x from both sides: 3x² - 9x - 3 = -9
- Add 9 to both sides: 3x² - 9x + 6 = 0
Step 3: Solve the Quadratic Equation
We now have a quadratic equation in the standard form (ax² + bx + c = 0). We can solve this using the quadratic formula:
- Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a
- In our equation: a = 3, b = -9, c = 6
Plugging these values into the quadratic formula:
x = (9 ± √((-9)² - 4 * 3 * 6)) / (2 * 3) x = (9 ± √(81 - 72)) / 6 x = (9 ± √9) / 6 x = (9 ± 3) / 6
This gives us two possible solutions:
- x1 = (9 + 3) / 6 = 2
- x2 = (9 - 3) / 6 = 1
Conclusion
Therefore, the solutions to the equation (2x+1)(x-3) = x(4-x) - 9 are x = 2 and x = 1.