Answer:
The solution of the equation is [tex]x=\frac{7\pm 3i\sqrt{3}}{2},\frac{7\pm 3i\sqrt{3}}{2}[/tex]
Step-by-step explanation:
Given : Equation [tex](x -5)^2 + 3(x -5) + 9 = 0[/tex]
To find : What is the solution of the equation?
Solution :
We have given the expression in quadratic form [tex]ax^2+bx+c=0[/tex]
Let [tex](x-5)=u[/tex] .....(10
The equation form is [tex]u^2 + 3u + 9 = 0[/tex]
The solution of a quadratic formula is, [tex]x=\frac{-b\pm\sqrt{b^2-4ac}} {2a}[/tex]
On comparing with general form,
a=1 ,b=3, c=9
Substitute in the formula,
[tex]u=\frac{-3\pm\sqrt{3^2-4(1)(9)}}{2(1)}[/tex]
[tex]u=\frac{-3\pm\sqrt{9-36}}{2}[/tex]
[tex]u=\frac{-3\pm\sqrt{-27}}{2}[/tex]
[tex]u=\frac{-3\pm 3\sqrt{3}i}{2}[/tex]
[tex]u_1=\frac{-3+3\sqrt{3}i}{2}[/tex]
[tex]u_2=\frac{-3-3\sqrt{3}i}{2}[/tex]
Substitute in equation (1),
[tex]x-5=u_1[/tex]
[tex]x-5=\frac{-3+3\sqrt{3}i}{2}[/tex]
[tex]x=\frac{-3+3\sqrt{3}i}{2}+5[/tex]
[tex]x=\frac{7\pm 3i\sqrt{3}}{2}[/tex]
[tex]x-5=u_2[/tex]
[tex]x-5=\frac{-3-3\sqrt{3}i}{2}[/tex]
[tex]x=\frac{-3-3\sqrt{3}i}{2}+5[/tex]
[tex]x=\frac{7\pm 3i\sqrt{3}}{2}[/tex]
Therefore, The solution of the equation is [tex]x=\frac{7\pm 3i\sqrt{3}}{2},\frac{7\pm 3i\sqrt{3}}{2}[/tex]
