4.2. Surd Equations 125
Exercises
- Consider the equation &??? + m = 1.
(a) As part of a strategy to obtain an equation without surds, derive
an equation in which the two surds are on opposite sides of the
equal sign.
(b) Square the two sides of the equation in (a) to obtain a new
equation.
(c) Rewrite the equation in (b) so that only the surd term appears
on one side.
(d) Derive a final equation by squaring the two sides of the equation
in (c).
(e) Show that, if the given equation has a solution, then x = 7.
(f) Verify that x = 7 does not satisfy the given equation.
(g) Check which of the equations (d), (c), (b), (a) is satisfied by
2 = 7.
(h) Consider the given equation along with the equations in (a),
(b), (c), (d). For each adjacent pair of equations, state which of
the pair implies the other. Explain how the extraneous solution
2 = 7 arises.
(i) Make up an equation similar to the given equation for which
2 = 7 is a solution.- Show how to solve a surd equation of the form
kdiii?ii+cx+d=Owhere k, a, b, c, d are real and ak # 0, by making the substitution
y = Jm. Determine conditions under which it has 0, 1, 2 real
roots.- Solve the equations
(a) J&T= x-2
(b) 4-=x+4
(c) 4&-Ti+x+4=0.- Solve the equations
(a) &TT = 3x - 5
(b) dm = 5 - 3x.