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(2x+1)^2-(x+13)=3x^2-2x+2

2 min read Jun 16, 2024
(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

  1. Expand the square:
    (2x+1)^2 = (2x+1)(2x+1) = 4x^2 + 4x + 1

  2. Rewrite the equation: 4x^2 + 4x + 1 - (x + 13) = 3x^2 - 2x + 2

  3. Distribute the negative sign: 4x^2 + 4x + 1 - x - 13 = 3x^2 - 2x + 2

  4. 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.

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