Curve and line

The equation of a curve C is y=2x² -8x+9 and the equation of a line L is x+ y=3

(1) Find the x co-ordinates of the points of intersection of L and C.
(2) Show that one of these points is also the stationary point of C?

Correct result:

a₁ =  2
a₂ =  1.5
b =  2

Solution:

$$\begin{gathered} 3-x=2\ast x^2-8\ast x+9 \\ 3-x=2\cdot x^2-8\cdot x+9 \\ -2x^2+7x-6=0 \\ 2x^2-7x+6=0 \\ a=2;b=-7;c=6 \\ D=b^2-4ac=7^2-4\cdot 2\cdot 6=1 \\ D>0 \\ {x}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{7\pm \sqrt{1}}{4}} \\ {x}_{1,2}={\displaystyle \frac{7\pm 1}{4}} \\ {x}_{1,2}=1.75\pm 0.25 \\ {x}_1=2 \\ {x}_2=1.5 \\ \text{ Factored form of the equation: } \\ 2(x-2)(x-1.5)=0 \\ {a}_1=x_1=2 \end{gathered}$$

Our quadratic equation calculator calculates it.

$a_{2}=x_{2}=1.5=\dfrac{ 3 }{ 2 }$

$4b-8=0 \ \\ 4b=8 \ \\ b=8 / 4=2 \ \\ =2$

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