Expanding the left side of the equation:
(x+5)^2 + (x-7)(x+7)= (x+5)(x+5) + (x^2 + 7x - 7x - 49)= x^2 + 5x + 5x + 25 + x^2 - 49= 2x^2 + 10x - 24
Now the equation becomes:
2x^2 + 10x - 24 = 6x - 19
Subtracting 6x and adding 19 to both sides:
2x^2 + 4x - 5 = 0
This is a quadratic equation that can be factored as:
(2x - 5)(x + 1) = 0
Setting each factor to zero:
2x - 5 = 02x = 5x = 5/2
x + 1 = 0x = -1
Therefore, the solutions are x = 5/2 and x = -1.
Expanding the left side of the equation:
(x+5)^2 + (x-7)(x+7)
= (x+5)(x+5) + (x^2 + 7x - 7x - 49)
= x^2 + 5x + 5x + 25 + x^2 - 49
= 2x^2 + 10x - 24
Now the equation becomes:
2x^2 + 10x - 24 = 6x - 19
Subtracting 6x and adding 19 to both sides:
2x^2 + 4x - 5 = 0
This is a quadratic equation that can be factored as:
(2x - 5)(x + 1) = 0
Setting each factor to zero:
2x - 5 = 0
2x = 5
x = 5/2
x + 1 = 0
x = -1
Therefore, the solutions are x = 5/2 and x = -1.