Find the constant.Find the constant k so that the quadratic equation 2x2 + 5x - k = 0 has two real solutions.
A quadratic equation ax^2 + bx + c = 0 has two real roots if b^2 > 4ac
For 2x^2 + 5x - k = 0 to have two real roots:
5^2 > 4*2*(-k)
=> 25 > -8k
=> k > -25/8
The term k can take any value greater than -25/8
When a quadratic has 2 real solutions, it's discriminant delta is positive or equal to zero.
delta > = 0
delta = b^2 - 4ac, where a,b,c are the coefficients of the quadratic.
Since it is not specified if the roots are different or they are equal, we'll consider both cases:
delta > 0
b^2 - 4ac > 0
We'll identify a,b,c:
2x^2 + 5x - k = 0
a = 2 , b = 5 , c = -k
delta = 25 + 8k
25 + 8k > 0
8k > -25
k > -25/8
Therefore, if the values of k are in the interval (-25/8 ; +infinite), the quadratic has 2 different real roots.
For delta = 0 => x1 = x2
25 + 8k = 0
8k = -25
k = -25/8
If k = -25/8, the roots of the quadratic are equal and real.