Intersection of Q2 with 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-coordinates of the points of intersection of L and C.
(ii) show that one of these points is also the

Correct answer:

x₁ =  2
x₂ =  1.5
s =  2

Step-by-step explanation:

$$\begin{gathered} C:y=2x²-8x+9 \\ L:x+y=3 \\ L:y=3-x \\ {y}_1=y_2 \\ 2\cdot x^2-8\cdot x+9=3-x \\ 2\cdot x^2-8\cdot x+9=3-x \\ 2x^2-7x+6=0 \\ a=2;b=-7;c=6 \\ D=b^2-4ac=7^2-4\cdot 2\cdot 6=1 \\ D>0 \\ {x}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{7\pm \sqrt{1}}{4} \\ {x}_{1,2}=\frac{7\pm 1}{4} \\ {x}_{1,2}=1.75\pm 0.25 \\ {x}_1=2 \\ {x}_2=1.5 \end{gathered}$$

Our quadratic equation calculator calculates it.

$$\begin{gathered} C^{\prime }:f^{\prime }(x)=d_y/d_x=4x-8 \\ {f}^{\prime }(s)=0 \\ 4s-8=0 \\ 4\cdot s-8=0 \\ 4s=8 \\ s=\frac{8}{4}=2 \\ s=2 \end{gathered}$$

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