1.
We need to find the roots of the next quadratic function:
[tex]z^2+8z+65[/tex]
Use the quadratic formula, which is given by the form ax²+bx+c:
[tex]\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]
Replace the values using a=1, b=8 and c=65
[tex]\frac{-8\pm\sqrt[]{(8)^2-4(1)(65)}}{2(1)}[/tex][tex]\frac{-8\pm\sqrt[]{64-260}}{2}=\frac{-8\pm\sqrt[]{-196}}{2}=\frac{-8\pm14i}{2}[/tex]
Simplify the expression:
[tex]\frac{-8\pm14i}{2}=\frac{2(-4\pm7i)}{2}=-4\pm7i[/tex]
Therefore, the roots of the given expression are:
x = -4+7i
x= -4 - 7i
2.
We need to find the quadratic using the integer coefficients:
We have that the root are :
[tex]\frac{1}{2}\pm\frac{3}{2}i[/tex]
Use the quadratic form ax²+bx+c
The roots are the solution for the quadratic formula, so:
[tex]\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}=\frac{1}{2}\pm\frac{3}{2}i[/tex]
Then:
[tex]\frac{1\pm\sqrt[]{9i^2}}{2}=\frac{1\pm\sqrt[]{-9}}{2}[/tex]
So we have that 2a =2. then a =2/2
a=1
Also, -b = 1
b = -1
b²-4ac = -9
(-1)²-4(1)c = -9
-4(1)c = -9 -1
-4c = -10
Solve for c
c = -10/-4
c = 5/2
Therefore, with these values, we can replace the quadratic form and we will find the result
a=1
b = -1
c = 5/2
[tex]ax^2+bx+c\text{ =0}[/tex][tex]x^2-x+\frac{5}{2}=0[/tex]
So, the last quadratic is the result.
