(2x%2B3)-(x-2)(x%2B1)%3D7.jpeg "(2x-3)(2x+3)-(x-2)(x+1)=7")
Solving the Equation: (2x-3)(2x+3)-(x-2)(x+1)=7
This article will guide you through solving the equation (2x-3)(2x+3)-(x-2)(x+1)=7. We will use algebraic manipulation to simplify the equation and ultimately find the value of 'x'.
Step 1: Expanding the Products
First, we need to expand the products on the left-hand side of the equation using the FOIL method (First, Outer, Inner, Last).
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(2x-3)(2x+3):
- First: (2x)(2x) = 4x²
- Outer: (2x)(3) = 6x
- Inner: (-3)(2x) = -6x
- Last: (-3)(3) = -9
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(x-2)(x+1):
- First: (x)(x) = x²
- Outer: (x)(1) = x
- Inner: (-2)(x) = -2x
- Last: (-2)(1) = -2
Step 2: Simplifying the Equation
Now, let's substitute the expanded products back into the equation:
4x² + 6x - 6x - 9 - (x² + x - 2x - 2) = 7
Simplifying further:
4x² - 9 - x² - x + 2x + 2 = 7
Combining like terms:
3x² + x - 7 = 7
Step 3: Rearranging the Equation
To solve for 'x', we need to bring all the terms to one side of the equation:
3x² + x - 14 = 0
Step 4: Solving the Quadratic Equation
The equation is now a quadratic equation in the form ax² + bx + c = 0. We can solve this using the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
In this case, a = 3, b = 1, and c = -14. Substituting these values into the quadratic formula:
x = (-1 ± √(1² - 4 * 3 * -14)) / (2 * 3)
x = (-1 ± √(169)) / 6
x = (-1 ± 13) / 6
Step 5: Finding the Solutions
Therefore, we get two possible solutions for 'x':
- x = (-1 + 13) / 6 = 2
- x = (-1 - 13) / 6 = -2.33 (approximately)
Conclusion
The solutions to the equation (2x-3)(2x+3)-(x-2)(x+1)=7 are x = 2 and x = -2.33. You can verify these solutions by substituting them back into the original equation.