To solve this equation, we first need to expand and simplify both sides:
(3x - 2)(x + 4) = (x - 5)² + 11
Expanding the left side:
3x^2 + 12x - 2x - 8 = (x - 5)(x - 5) + 113x^2 + 10x - 8 = (x^2 - 10x + 25) + 113x^2 + 10x - 8 = x^2 - 10x + 25 + 113x^2 + 10x - 8 = x^2 - 10x + 36
Now we can combine like terms:
2x^2 + 20x - 44 = 0
To solve the quadratic equation, we can either factor it or use the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
Plugging in the values of a=2, b=20, and c=-44:
x = (-20 ± √((20)² - 42(-44))) / 22x = (-20 ± √(400 + 352)) / 4x = (-20 ± √752) / 4x = (-20 ± √(1647)) / 4x = (-20 ± 4√47) / 4x = -5 ± √47
Therefore, the solutions to the equation are x = -5 + √47 and x = -5 - √47.
To solve this equation, we first need to expand and simplify both sides:
(3x - 2)(x + 4) = (x - 5)² + 11
Expanding the left side:
3x^2 + 12x - 2x - 8 = (x - 5)(x - 5) + 11
3x^2 + 10x - 8 = (x^2 - 10x + 25) + 11
3x^2 + 10x - 8 = x^2 - 10x + 25 + 11
3x^2 + 10x - 8 = x^2 - 10x + 36
Now we can combine like terms:
2x^2 + 20x - 44 = 0
To solve the quadratic equation, we can either factor it or use the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
Plugging in the values of a=2, b=20, and c=-44:
x = (-20 ± √((20)² - 42(-44))) / 22
x = (-20 ± √(400 + 352)) / 4
x = (-20 ± √752) / 4
x = (-20 ± √(1647)) / 4
x = (-20 ± 4√47) / 4
x = -5 ± √47
Therefore, the solutions to the equation are x = -5 + √47 and x = -5 - √47.