3.2 Equations with roots
From Förberedande kurs i matematik 1
| Theory | Exercises |
Contents:
- Equations of the type \displaystyle \sqrt{ax+b}= cx +d
- Spurious roots
Learning outcomes
After this section, you will have learned to:
- Solve simple equations containing roots.
- Manage spurious roots, and know when they might appear.
Equations with roots
There are many different types of equations containing roots. Some examples are
| \displaystyle \sqrt{x} + 3x = 2\,, |
| \displaystyle \sqrt{x - 1} - 2x = x^2\,, |
| \displaystyle \sqrt[\scriptstyle3]{x + 2} = x\,\mbox{.} |
To solve equations with roots we need to get rid of the root sign. The strategy is to reformulate the equation so that the root sign only appears on one side of the equals sign. Then we can square both sides of the equation (in the case of quadratic roots), so that the root sign disappears, and solve the resulting (squared) equation. We have to be careful though, since a solution to the resulting equation might not be a solution to the original equation. We lose information when squaring, since both positive and negative quantities become positive after squaring. Therefore, upon squaring the equation, we may generate extra roots which solve the squared equation but not the original equation. These extra roots are called spurious roots. Thus we must examine the solutions that appear, and identify the spurious roots by checking whether or not they solve the original equation.
Example 1
Consider a simple (trivial) equation
| \displaystyle x = 2\mbox{.} |
If we square both sides of this equation, we get
| \displaystyle x^2 = 4\mbox{.} |
This new equation has two solutions \displaystyle x = 2 or \displaystyle x = -2. The solution \displaystyle x = 2 satisfies the original equation, while \displaystyle x = -2 only satisfies the squared equation. So we see that the squared equation has more solutions than the original equation. In this case \displaystyle x = -2 is a spurious root.
Example 2
Solve the equation \displaystyle \ 2\sqrt{x - 1} = 1 - x.
If we square both sides of the equation we get
| \displaystyle 4(x - 1) = (1 - x)^2. |
Expanding the square on the right-hand side we see that
| \displaystyle 4(x - 1)= 1 - 2x + x^2\,\mbox{.} |
This is a quadratic equation, which can be written as
| \displaystyle x^2 - 6x + 5 = 0\,\mbox{.} |
This can be solved by completing the square or by using the general solution formula. Either way the solutions are \displaystyle x = 3 \pm 2, i.e. \displaystyle x = 1 or \displaystyle x = 5.
Since we squared the original equation, there is a risk that spurious roots have been introduced, and therefore we need to check whether \displaystyle x=1 and \displaystyle x=5 are also solutions to the original equation:
- \displaystyle x = 1 gives that \displaystyle \mbox{LHS} = 2\sqrt{1 - 1} = 0 and \displaystyle \mbox{RHS} = 1 - 1 = 0. So \displaystyle \mbox{LHS} = \mbox{RHS} and the equation is satisfied!
- \displaystyle x = 5 gives that \displaystyle \mbox{LHS} = 2\sqrt{5 - 1} = 2\cdot2 = 4 and \displaystyle \mbox{RHS} = 1 - 5 = -4. So \displaystyle \mbox{LHS} \ne \mbox{RHS} and the equation is not satisfied!
Thus the equation has only one solution \displaystyle x = 1.
![[Image]](/wikis/2008/bridgecourse1-ImperialCollege/img_auth.php/metapost/5/b/7/5b71bb3f756c81b52a57bb91f01acf60.png)
Study advice
The basic and final tests
After you have read the text and worked through the exercises, you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.
Keep in mind that...
When squaring an equation remember that the solutions obtained might not be the solutions to the original equation i.e. we might generate spurious roots. This is because we lose information when squaring, since minus signs disappear. Therefore, one must verify that the solutions obtained are indeed solutions to the original equation.
You should always test the solution in the original equation.
Reviews
For those of you who want to deepen your understanding or need more detailed explanations consider the following reference:
Understanding Algebra - English online book for pre-university studies
Useful web sites
Webmath.com can help you to simplify root expressions.
