# 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 1Find all real solutions to the equation.^{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 2Find all real solutions to the equation.^{2} = 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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