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Solving the Equation: (x-2)² + x(x-3) = 3(x+4)(x-3) - (x+2)(x-1) + 2
This article will guide you through the process of solving the given equation:
(x-2)² + x(x-3) = 3(x+4)(x-3) - (x+2)(x-1) + 2
Let's break down the steps involved:
1. Expanding the Expressions
First, we need to expand all the expressions in the equation to simplify it.
- (x-2)²: This is a squared binomial. We can expand it using the formula (a-b)² = a² - 2ab + b². So, (x-2)² = x² - 4x + 4.
- x(x-3): This is a simple multiplication. x(x-3) = x² - 3x.
- 3(x+4)(x-3): This is a multiplication of three terms. We can expand it in two steps:
- First, expand (x+4)(x-3) using the distributive property or the FOIL method: (x+4)(x-3) = x² + x - 12.
- Then, multiply the result by 3: 3(x² + x - 12) = 3x² + 3x - 36.
- -(x+2)(x-1): This involves multiplying and distributing a negative sign.
- Expand (x+2)(x-1) using the distributive property or the FOIL method: (x+2)(x-1) = x² + x - 2.
- Then, multiply the result by -1: -(x² + x - 2) = -x² - x + 2.
- (x+2)(x-1): This is a multiplication of two binomials. We can expand it using the distributive property or the FOIL method: (x+2)(x-1) = x² + x - 2.
Now, our equation looks like this:
x² - 4x + 4 + x² - 3x = 3x² + 3x - 36 - x² - x + 2 + 2
2. Combining Like Terms
Next, we will combine the terms with the same power of x on both sides of the equation.
2x² - 7x + 4 = 2x² + x - 32
3. Isolating x
To solve for x, we need to get all the x terms on one side of the equation and the constant terms on the other side.
- Subtract 2x² from both sides: -7x + 4 = x - 32
- Subtract x from both sides: -8x + 4 = -32
- Subtract 4 from both sides: -8x = -36
4. Solving for x
Finally, we can isolate x by dividing both sides by -8:
x = 4.5
Therefore, the solution to the equation (x-2)² + x(x-3) = 3(x+4)(x-3) - (x+2)(x-1) + 2 is x = 4.5.