Answer:
Part 1) The x-component of the vertex is 2 and the y-component of the vertex is -18
Part 2) The discriminant is 144
Step-by-step explanation:
we have
[tex]f(x)=2x^{2}-8x-10[/tex]
step 1
Find the discriminant
The discriminant of a quadratic equation is equal to
[tex]D=b^{2}-4ac[/tex]
in this problem we have
[tex]f(x)=2x^{2}-8x-10[/tex]
so
[tex]a=2\\b=-8\\c=-10[/tex]
substitute
[tex]D=(-8)^{2}-4(2)(-10)[/tex]
[tex]D=64+80=144[/tex]
The discriminant is greater than zero, therefore the quadratic equation has two real solutions
step 2
Find the vertex
Convert the quadratic equation into vertex form
[tex]f(x)+10=2x^{2}-8x[/tex]
[tex]f(x)+10=2(x^{2}-4x)[/tex]
[tex]f(x)+10+8=2(x^{2}-4x+4)[/tex]
[tex]f(x)+18=2(x-2)^{2}[/tex]
[tex]f(x)=2(x-2)^{2}-18[/tex] -----> equation in vertex form
The vertex is the point (2,-18)
therefore
The x-component of the vertex is 2
The y-component of the vertex is -18
