Find all solutions of the equation |x^2 - 30x - 1| + |x^2 - 30x + 1| = 30.
To find all solutions of the equation |x^2 - 30x - 1| + |x^2 - 30x + 1| = 30, let's break it down step by step:
Step 1: Simplify the absolute value expressions
To simplify the equation, we need to consider two cases: when the expressions inside the absolute value signs are positive, and when they are negative.
Case 1: When x^2 - 30x - 1 is positive
In this case, the equation becomes:
(x^2 - 30x - 1) + (x^2 - 30x + 1) = 30
Simplifying further gives us:
2x^2 - 60x = 30
Step 2: Solve for x in the positive case
To solve the equation, we can rewrite it in standard quadratic form:
2x^2 - 60x - 30 = 0
Dividing the whole equation by 2 gives us:
x^2 - 30x - 15 = 0
We can now solve this quadratic equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
From the equation above, we can see that a = 1, b = -30, and c = -15. Substituting these values into the quadratic formula, we get:
x = (-(-30) ± √((-30)^2 - 4(1)(-15))) / (2(1))
x = (30 ± √(900 + 60)) / 2
x = (30 ± √960) / 2
x = (30 ± 4√60) / 2
x = 15 ± 2√60
Therefore, in the positive case, the solutions for x are x = 15 + 2√60 and x = 15 - 2√60.
Case 2: When x^2 - 30x - 1 is negative
In this case, the equation becomes:
-(x^2 - 30x - 1) + (x^2 - 30x + 1) = 30
Simplifying further gives us:
-2x^2 + 60x = 30
Step 3: Solve for x in the negative case
Similar to the positive case, we need to rewrite the equation in standard quadratic form:
-2x^2 + 60x - 30 = 0
Dividing the whole equation by -2 gives us:
x^2 - 30x + 15 = 0
Using the quadratic formula again, we can solve this equation:
x = (-(-30) ± √((-30)^2 - 4(1)(15))) / (2(1))
x = (30 ± √(900 - 60)) / 2
x = (30 ± √840) / 2
x = (30 ± 2√210) / 2
x = 15 ± √210
Therefore, in the negative case, the solutions for x are x = 15 + √210 and x = 15 - √210.
Step 4: Combine the solutions from both cases
The solutions for x in both the positive and negative cases are:
Positive case: x = 15 + 2√60 and x = 15 - 2√60
Negative case: x = 15 + √210 and x = 15 - √210
So, the complete set of solutions for the equation |x^2 - 30x - 1| + |x^2 - 30x + 1| = 30 is:
x = 15 + 2√60, x = 15 - 2√60, x = 15 + √210, and x = 15 - √210.