%5E2-(x%2B13)%3D3x%5E2-2x%2B2.jpeg "(2x+1)^2-(x+13)=3x^2-2x+2")
Solving the Equation: (2x+1)^2-(x+13)=3x^2-2x+2
This article will guide you through the process of solving the equation (2x+1)^2-(x+13)=3x^2-2x+2.
Expanding and Simplifying
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Expand the square:
(2x+1)^2 = (2x+1)(2x+1) = 4x^2 + 4x + 1 -
Rewrite the equation: 4x^2 + 4x + 1 - (x + 13) = 3x^2 - 2x + 2
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Distribute the negative sign: 4x^2 + 4x + 1 - x - 13 = 3x^2 - 2x + 2
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Combine like terms: x^2 + 7x - 14 = 0
Solving the Quadratic Equation
Now we have a quadratic equation in standard form (ax^2 + bx + c = 0). There are several methods to solve this equation:
1. Factoring:
- Find two numbers that add up to 7 and multiply to -14. These numbers are 14 and -1.
- Rewrite the equation: (x + 14)(x - 1) = 0
- Set each factor equal to zero: x + 14 = 0 or x - 1 = 0
- Solve for x: x = -14 or x = 1
2. Quadratic Formula:
- The quadratic formula solves for x in any quadratic equation: x = (-b ± √(b^2 - 4ac)) / 2a
- Substitute the values from our equation (a = 1, b = 7, c = -14): x = (-7 ± √(7^2 - 4 * 1 * -14)) / (2 * 1)
- Simplify: x = (-7 ± √(81)) / 2 x = (-7 ± 9) / 2
- Solve for x: x = 1 or x = -14
Solution
Therefore, the solutions to the equation (2x+1)^2-(x+13)=3x^2-2x+2 are x = 1 and x = -14.