Surd or Radical Equations

To solve an equation with a surd

4312 x First isolate the surd

This means to get any terms not under the square root on the other side of the equal sign

+ 3 + 3

712 x2 2

You must square the whole side NOT each term. A square "undoes" or cancels a square root4912 x

Now square both sides

Now solve for x

+ 1 + 1

502 x25x

You MUST check this answerSince you squared both sides of the equation, negatives disappear. It is possible to get an answer that doesn't work when you plug it back in

431252 44 It checks!

Let's try another one:

0112 3

1

x First isolate the surd

- 1

112 3

1

x3 3

112 x

Now since it is a 1/3 power this means the same as a cube root so cube both sides

Now solve for x

- 1 - 1

22 x1x

Let's check this answer

011123 00 It checks!

Remember that the 1/3 power means the same thing as a cube root.

- 1

One more to see extraneous solution:

313 xx The surd is already isolated

2 2You must square the whole side

NOT each term.

9613 2 xxx

Square both sides

Since you have a quadratic equation (has an x2 term) get everything on one side = 0 and see if you can factor this

1,8 xx

You MUST check these answers

55

38183

313 xxThis must be FOILed

0892 xx

018 xx 22

31113

Doesn't work!Extraneous

It checks!

a solution that you find algebraically but DOES NOT make a true statement when you substitute it back into the equation.

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au