ANSWER
[tex]x = \frac{3}{2} \: or \: x = - 3[/tex]
EXPLANATION
We want to find the roots of the parabola with equation:
[tex]2 {x}^{2} + 5x - 9 = 2x[/tex]
We need to write this in the standard quadratic equation form.
We group all terms on the left to get:
[tex]2 {x}^{2} + 5x - 2x - 9 = 0[/tex]
We simplify to get:
[tex]2 {x}^{2} +3x- 9 = 0[/tex]
We now compare to:
[tex]a {x}^{2} + bx + c = 0[/tex]
[tex] \implies \: a = 2 , \: \: b = 3 \: \: and \: c=- 9[/tex]
[tex] \implies ac = 2 \times - 9 = - 18[/tex]
The factors of -18 that sums up to 3 are -3, 6.
We split the middle term with these factors to get:
[tex]2 {x}^{2} +6x - 3x- 9 = 0[/tex]
Factor by grouping:
[tex]2x(x + 3) -3(x + 3) = 0[/tex]
Factor again to obtain:
[tex](2x - 3)(x + 3) = 0[/tex]
Apply the zero product principle to get:
[tex]2x - 3 = 0 \: or \: x + 3 = 0[/tex]
[tex] \implies \: x = \frac{3}{2} \: or \: x = - 3[/tex]
