Answers to Matched Exercises of Equations of the Quadratic Form

Answers to the matched exercises in Tutorial on Equations of the Quadratic Form are presented.

Matched Exercise 1

Find all real solutions to the equation.
x 4 - 2 x 2 - 3 = 0

Let u = x
2
The above equation may be written as
u
2 - 2 u - 3 = 0
Solve the above for u to obtain the solutions
u = - 1 and u = 3
We now solve for x.
u = x
2 = - 1 , this equation has no real solutions.
u = x
2 = 3 gives x = √3 and x = - √3
The given equation has 2 real solutions.
x3 = √3
x4 = - √3

Matched Exercise 2

Find all real solutions to the equation.
x - 3 √x - 4 = 0
Let u = √x which gives u2 = x
The given equation is now written in terms of u as follows
u
2 - 3 u - 4 = 0
Solve the above quadratic equation for u to obtain
u = - 1 and u = 4
Solve for x
u = √x = - 1 , this equation has no real solution √x is positive
u = √x = 4 , square both sides to obtain the solution
x = 16

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