Question

If r is a constant such that \(|x^2-4x-13| = r\) has exactly three distinct real roots, then the value of r is

• A. 15
• B. 18
• C. 17
• D. 21

Answer 1

The Correct Option is C

Given the equation \(|x^2 - 4x - 13| = r\)
The equation \(x^2 - 4x - 13 = r\) or \(x^2 - 4x - 13 = -r\) should have exactly three distinct real roots combined.
1. Let's solve for the equation \(x^2 - 4x - 13 = r\)
\(x^2 - 4x - 13 - r = 0\)
Using the discriminant \(b^2 - 4ac\) of a quadratic equation \(ax^2 + bx + c\), for the equation to have real roots, \(b^2 - 4ac \geq 0\).
So, for our equation: 
\(16 - 4(1)(-13 - r) \geq 0\) 
\(16 + 52 + 4r \geq 0\) 
\(68 + 4r \geq 0\)
\(r \leq -17\)
2. Let's solve for the equation 
\(x^2 - 4x - 13 = -r\) 
\(x^2 - 4x - 13 + r = 0\)
Again, using the discriminant: 
\(16 - 4(1)(-13 + r) \geq 0\)
\(16 + 52 - 4r \geq 0\) 
\(68 - 4r \geq 0\)
\(r \leq 17\)
Since we need a total of three real roots for both equations combined and considering that for \(r \leq -17\) the first equation will give real roots and the second equation will not, the only possible value from the options that can provide exactly 3 real roots is \(r = 17\).

Therefore, the correct option is (C): 17

Answer 2

There are precisely three different real roots for\( |x^2 - 4x - 13| = r. \)
Now let's complete the quadratic equation's squares.
\(|(x - 2)^2 - 17| = r\)
\((x - 2)^2 - 17 = r\) and \((x - 2)^2 - 17 = -r\)
\((x - 2)^2 = r + 17\) and \((x - 2)^2 = -r + 17\)
As a result, for \(r = 17\), we obtain exactly three roots. 
The response is 17.