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Solving the Equation: (5x-9)(x-1)-x(x-2)=0
This article will guide you through the process of solving the given quadratic equation: (5x-9)(x-1)-x(x-2)=0.
Step 1: Expand the Equation
First, we need to expand the equation by multiplying out the brackets:
- (5x-9)(x-1) = 5x² - 5x - 9x + 9 = 5x² - 14x + 9
- x(x-2) = x² - 2x
Now our equation looks like this: 5x² - 14x + 9 - (x² - 2x) = 0
Step 2: Simplify the Equation
Next, we simplify the equation by combining like terms:
- 5x² - x² - 14x + 2x + 9 = 0
- 4x² - 12x + 9 = 0
Step 3: Solve for x
We can now solve for x using the quadratic formula. The quadratic formula is:
x = (-b ± √(b² - 4ac)) / 2a
Where:
- a = 4
- b = -12
- c = 9
Substituting these values into the quadratic formula, we get:
x = (12 ± √((-12)² - 4 * 4 * 9)) / (2 * 4)
x = (12 ± √(144 - 144)) / 8
x = (12 ± √0) / 8
x = 12 / 8
x = 3/2
Therefore, the solution to the equation (5x-9)(x-1)-x(x-2)=0 is x = 3/2.
Conclusion
By following the steps outlined above, we successfully solved the quadratic equation. Remember, the quadratic formula is a powerful tool for solving equations of this type. Always ensure you simplify and expand the equation correctly to avoid errors.