For what value of k does the equation x2 + k = kx – 8 have:
Two distinct real roots
One real root
No real root
To find the values of k that result in two distinct real roots, one real root, or no real root for the equation x^2 + k = kx - 8, we need to use the quadratic formula. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, b = -k, a = 1, and c = -k - 8.
1. Two distinct real roots:
For the equation to have two distinct real roots, the discriminant, which is the expression under the square root in the quadratic formula, must be greater than 0. So we have:
(b^2 - 4ac) > 0
(-k)^2 - 4(1)(-k - 8) > 0
k^2 + 4k + 32 > 0
Solving this quadratic inequality, we can find the range of k values that satisfy it.
2. One real root:
For the equation to have one real root, the discriminant must be equal to 0. So we have:
(b^2 - 4ac) = 0
(-k)^2 - 4(1)(-k - 8) = 0
k^2 + 4k + 32 = 0
Solving this quadratic equation, we can find the values of k that satisfy it.
3. No real root:
For the equation to have no real root, the discriminant must be less than 0. So we have:
(b^2 - 4ac) < 0
(-k)^2 - 4(1)(-k - 8) < 0
k^2 + 4k + 32 < 0
Solving this quadratic inequality, we can find the range of k values that satisfy it.
By solving the above quadratic inequalities and equations, we can determine the values of k that lead to two distinct real roots, one real root, or no real root for the given equation.