To determine the values of [tex]\(k\)[/tex] for which the quadratic equation
[tex]\[
(k+4)z^2 + 7z - 8 = 0
\][/tex]
has two real solutions, we need to analyze the discriminant of the equation. Recall that a quadratic equation [tex]\(az^2 + bz + c = 0\)[/tex] will have two distinct real solutions if its discriminant [tex]\(\Delta\)[/tex] is greater than zero. The discriminant for a quadratic equation is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
In our equation, the coefficients are:
- [tex]\(a = k + 4\)[/tex]
- [tex]\(b = 7\)[/tex]
- [tex]\(c = -8\)[/tex]
Thus, the discriminant [tex]\(\Delta\)[/tex] for our quadratic equation is:
[tex]\[
\Delta = 7^2 - 4(k+4)(-8)
\][/tex]
Simplify the discriminant:
[tex]\[
\Delta = 49 - 4(k+4)(-8)
\][/tex]
[tex]\[
\Delta = 49 + 32(k + 4)
\][/tex]
[tex]\[
\Delta = 49 + 32k + 128
\][/tex]
[tex]\[
\Delta = 32k + 177
\][/tex]
For the quadratic equation to have two distinct real solutions, we need the discriminant to be greater than zero:
[tex]\[
32k + 177 > 0
\][/tex]
Solving this inequality for [tex]\(k\)[/tex], we get:
[tex]\[
32k > -177
\][/tex]
[tex]\[
k > -\frac{177}{32}
\][/tex]
Hence, the values of [tex]\(k\)[/tex] for which the given quadratic equation has two real solutions are:
[tex]\[
k > -\frac{177}{32}
\][/tex]
This can also be written as:
[tex]\[
k > -5.53125
\][/tex]
This means [tex]\(k\)[/tex] should be greater than [tex]\(-\frac{177}{32}\)[/tex]. This interval of solutions is:
[tex]\[
k \in \left( -\frac{177}{32}, \infty \right)
\][/tex]
