Write as a Vector Equality y=x-4 , y=-x^2+2x+4

Problem

y=x−4,y=−x2+2*x+4

Solution

1. Set the two expressions for y equal to each other to find the intersection points of the linear and quadratic functions.

x−4=−x2+2*x+4

2. Rearrange the equation into standard quadratic form a*x2+b*x+c=0 by moving all terms to the left side.

x2−x−8=0

3. Apply the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a) to solve for x

x=(−(−1)±√(,(−1)2−4*(1)*(−8)))/(2*(1))

4. Simplify the discriminant and the expression for x

x=(1±√(,1+32))/2

x=(1±√(,33))/2

5. Determine the corresponding y values by substituting the x values back into the linear equation y=x−4

y=(1±√(,33))/2−8/2

y=(−7±√(,33))/2

6. Express the system as a vector equality by equating the vector form of the two functions.

([x],[x−4])=([x],[−x2+2*x+4])

Final Answer

([x],[x−4])=([x],[−x2+2*x+4])

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