Final answer:
The real solutions to the quadratic equation x²-4x-24=0 are obtained using the quadratic formula, resulting in x = 2 + √7 and x = 2 - √7.
Step-by-step explanation:
To find all real solutions to the quadratic equation x²-4x-24=0, we can use the quadratic formula. This formula is applicable to any equation of the form ax²+bx+c=0. The quadratic formula is expressed as x = (-b ± √(b²-4ac))/(2a).
For the given equation, a=1, b=-4, and c=-24. Plugging these values into the quadratic formula, we get:
x = (4 ± √((4)²-4(1)(-24)))/(2(1))
x = (4 ± √(16+96))/(2)
x = (4 ± √(112))/(2)
x = (4 ± √(4√7))/(2)
x = (4 ± 2√7)/(2)
x = 2 ± √7
Thus, the real solutions to the equation are x = 2 + √7 and x = 2 - √7.