(3x+4)(3x-4)-(2x+5)^2=(x-5)^2+(2x+1)^2-(x^2-2x)+(x-1)^2

Jasonbradley

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Jun 16, 2024

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Solving the Equation: (3x+4)(3x-4)-(2x+5)^2=(x-5)^2+(2x+1)^2-(x^2-2x)+(x-1)^2

This article will guide you through the steps to solve the given equation:

(3x+4)(3x-4)-(2x+5)^2=(x-5)^2+(2x+1)^2-(x^2-2x)+(x-1)^2

Simplifying the Equation

1. Expand the products:

   • (3x+4)(3x-4): This is a difference of squares pattern: (a+b)(a-b) = a^2 - b^2. Therefore, (3x+4)(3x-4) = (3x)^2 - 4^2 = 9x^2 - 16.
   • (2x+5)^2: This is a perfect square trinomial: (a+b)^2 = a^2 + 2ab + b^2. So, (2x+5)^2 = (2x)^2 + 2(2x)(5) + 5^2 = 4x^2 + 20x + 25.
   • (x-5)^2: Similar to above, (x-5)^2 = x^2 - 10x + 25.
   • (2x+1)^2: (2x+1)^2 = 4x^2 + 4x + 1.
   • (x-1)^2: (x-1)^2 = x^2 - 2x + 1.

2. Substitute the expanded expressions into the equation:

   9x^2 - 16 - (4x^2 + 20x + 25) = x^2 - 10x + 25 + 4x^2 + 4x + 1 - (x^2 - 2x) + x^2 - 2x + 1

3. Distribute the negative sign:

   9x^2 - 16 - 4x^2 - 20x - 25 = x^2 - 10x + 25 + 4x^2 + 4x + 1 - x^2 + 2x + x^2 - 2x + 1

Combining Like Terms

4. Combine terms with the same variable and exponent:

   (9x^2 - 4x^2 + x^2 - 4x^2 + x^2) + (-20x - 10x + 4x + 2x - 2x) + (-16 - 25 + 25 + 1 + 1 + 1) = 0

5. Simplify:

   3x^2 - 26x - 14 = 0

Solving the Quadratic Equation

6. Factor the quadratic equation:

   This equation is not easily factorable. Therefore, we can use the quadratic formula to find the solutions for x:

   x = (-b ± √(b^2 - 4ac)) / 2a

   where a = 3, b = -26, and c = -14.

7. Substitute the values into the quadratic formula and solve:

   x = (26 ± √((-26)^2 - 4 * 3 * -14)) / (2 * 3) x = (26 ± √(676 + 168)) / 6 x = (26 ± √844) / 6 x = (26 ± 2√211) / 6

8. Simplify the solutions:

   x = (13 ± √211) / 3

Conclusion

Therefore, the solutions to the equation (3x+4)(3x-4)-(2x+5)^2=(x-5)^2+(2x+1)^2-(x^2-2x)+(x-1)^2 are x = (13 + √211) / 3 and x = (13 - √211) / 3.