How to solve parametric quadratic function?

Question

i have problem to solve following equation:

Find $b$ and $c$ if the graph of the function $y = 2x ^ 2 + bx + c$ touches the axis $Ox$ at the point $(1; 0)$.

Can anyone help me?

Answer 1

$y = 2x ^ 2 + bx + c$ passes through the point $(1,0)$

$2+b+c=0\tag{1}$

Furthermore it is tangent to $x$-axis, so the solution $x=1$ must be double

Which means that the discriminant of $2x ^ 2 + bx + c=0$ is zero, that is

$b^2-8c=0\tag{2}$ From $(1)$ solve $c=-b-2$ and plug into $(2)$

$b^2-8(-b-2)=0\to (b+4)^2=0\to b=-4$ and $c=2$

Answer 2

A quadratic that touches the $x$ axis when $x=1$ has the form $y=a(x-1)^2=ax^2-2ax+a.$

Equate the coefficients to those of $2x^2+bx+c$ to find $b$ and $c$.