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(2x+3)+(x+5) ^2+ (2x+3)-(x+5) ^2=10x^2+92

4 min read Jun 16, 2024
(2x+3)+(x+5) ^2+ (2x+3)-(x+5) ^2=10x^2+92

Solving the Equation: (2x+3)+(x+5)^2 + (2x+3)-(x+5)^2 = 10x^2 + 92

This article will walk through the steps involved in solving the equation (2x+3)+(x+5)^2 + (2x+3)-(x+5)^2 = 10x^2 + 92.

Simplifying the Equation

First, we simplify the equation by expanding the squares and combining like terms:

  • Expand the squares: (x+5)^2 = x^2 + 10x + 25

  • Substitute the expanded terms back into the equation: (2x+3) + (x^2 + 10x + 25) + (2x+3) - (x^2 + 10x + 25) = 10x^2 + 92

  • Combine like terms: 2x + 3 + x^2 + 10x + 25 + 2x + 3 - x^2 - 10x - 25 = 10x^2 + 92 4x + 6 = 10x^2 + 92

Rearranging the Equation

Next, we rearrange the equation to set it equal to zero:

  • Subtract 4x and 6 from both sides: 0 = 10x^2 + 92 - 4x - 6 0 = 10x^2 - 4x + 86

Solving the Quadratic Equation

We now have a quadratic equation in the form of ax^2 + bx + c = 0. There are a few methods to solve quadratic equations:

  • Factoring: This method is used when the equation can be factored into two binomials. However, in this case, the equation cannot be easily factored.

  • Quadratic Formula: The quadratic formula is a general solution for quadratic equations:

    x = (-b ± √(b^2 - 4ac)) / 2a

    where a = 10, b = -4, and c = 86.

  • Completing the Square: This method involves manipulating the equation to create a perfect square trinomial. However, this method can be more complex for this particular equation.

Using the Quadratic Formula:

  1. Substitute the values of a, b, and c into the quadratic formula: x = (4 ± √((-4)^2 - 4 * 10 * 86)) / (2 * 10)

  2. Simplify the expression: x = (4 ± √(-3400)) / 20 x = (4 ± 2√850i) / 20 (where 'i' is the imaginary unit, √-1) x = (1 ± √212.5i) / 5

Therefore, the solutions to the equation (2x+3)+(x+5)^2 + (2x+3)-(x+5)^2 = 10x^2 + 92 are complex numbers: x = (1 + √212.5i) / 5 and x = (1 - √212.5i) / 5.

Conclusion

This equation is a good example of how to solve equations involving squares and rearranging them into a quadratic form. While the solution involves complex numbers, the process demonstrates the steps involved in manipulating and solving equations.

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