Write as a Vector Equality y=x+20 , y=x^2-4x+5

Problem

{[y=x+20],[y=x2−4*x+5])

Solution

1. Set the equations equal to find the intersection points of the two functions.

x2−4*x+5=x+20

2. Rearrange into standard quadratic form by subtracting x and 20 from both sides.

x2−5*x−15=0

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

x=(−(−5)±√(,(−5)2−4*(1)*(−15)))/(2*(1))

4. Simplify the discriminant and the expression for x

x=(5±√(,25+60))/2

x=(5±√(,85))/2

5. Determine the corresponding y-values by substituting the x values back into the linear equation y=x+20

y=(5±√(,85))/2+40/2

y=(45±√(,85))/2

6. Express as a vector equality by grouping the x and y components into a single vector equation representing the solution set.

[[x],[y]]=[[(5±√(,85))/2],[(45±√(,85))/2]]

Final Answer

[[x],[y]]=[[(5±√(,85))/2],[(45±√(,85))/2]]

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