Question

|x+4|=a^2+4 При каких значениях параметра уравнение имеет два корня?

Answer 1

Так как а^2+4>4 при любом "а" , то данное уравнение будет иметь два корня при а€(-~;+~)
(принадлежит от минус бесконечность до плюс бесконечность)

Answer 2

To find the values of the parameter 'a' for which the equation |x + 4| = a^2 + 4 has two real roots, we need to analyze the possibilities for the expression |x + 4|.

The absolute value function |x| is defined as follows:

• For x >= 0, |x| = x.
• For x < 0, |x| = -x.

Now let's apply this definition to our equation |x + 4| = a^2 + 4:

1. If (x + 4) >= 0, then |x + 4| = x + 4.
2. If (x + 4) < 0, then |x + 4| = -(x + 4) = -x - 4.

So we have two possibilities for the equation:

1. If (x + 4) >= 0, the equation becomes: x + 4 = a^2 + 4
2. If (x + 4) < 0, the equation becomes: -x - 4 = a^2 + 4

Now, let's solve each of these equations separately to find the values of 'a' that give us two real roots:

1. x + 4 = a^2 + 4 Subtract 4 from both sides: x = a^2 Now, for this equation to have real roots, 'a^2' must be greater than or equal to 0 (since any real number squared is non-negative).

2. -x - 4 = a^2 + 4 Add 4 to both sides: -x = a^2 + 8 Multiply by -1 (this is allowed because we know 'a^2 + 8' is positive): x = -(a^2 + 8) Again, for this equation to have real roots, 'a^2 + 8' must be greater than or equal to 0.

In summary, for the equation |x + 4| = a^2 + 4 to have two real roots, both 'a^2' and 'a^2 + 8' must be greater than or equal to 0.

So, the range of 'a' that satisfies this condition is: a^2 >= 0 and a^2 + 8 >= 0

a^2 >= 0 is true for all real values of 'a', because squares of real numbers are non-negative.

a^2 + 8 >= 0 can be rewritten as a^2 >= -8, which is again true for all real values of 'a'.

Therefore, the equation |x + 4| = a^2 + 4 has two real roots for all real values of 'a'.