The equations
[tex]2x^2+9x-5=0[/tex]
and
[tex]2\left(t-\dfrac12\right)^2+9\left(t-\dfrac12\right)-5=0[/tex]
are really the same, we're just setting [tex]x=t-\dfrac12[/tex], or [tex]t=x+\dfrac12[/tex].
[tex]2x^2+9x-5=(2x-1)(x+5)=0\implies x=\dfrac12,x=-5[/tex]
So we get
[tex]t=\dfrac12+\dfrac12=1,t=-5+\dfrac12=-\dfrac92[/tex]
