Final answer:
To find a positive real number such that its square is equal to 14 times the number increased by 392, we can use the quadratic equation. The positive real number is x = 28.
Step-by-step explanation:
To find a positive real number such that its square is equal to 14 times the number increased by 392, we can use the quadratic equation. Let's assume the number is x. The equation is x^2 = 14x + 392. Rearranging the equation, we get x^2 - 14x - 392 = 0. We can solve this quadratic equation using the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a).
Plugging in the values a=1, b=-14, and c=-392 into the quadratic formula, we get:
x = (-(-14) ± √((-14)^2 - 4(1)(-392)))/(2(1))
x = (14 ± √(196 + 1568))/2
x = (14 ± √1764)/2
x = (14 ± 42)/2
Therefore, the possible values of x are x = (14 + 42)/2 = 28, or x = (14 - 42)/2 = -14. However, since we are looking for a positive real number, the answer is x = 28.