Overview:
A radical equation is an equation that contains a radical that includes a variable. It is possible to solve many radical equations in two steps. However, it is important to check the solution in the original equation to make sure it checks, as it may not be a real solution.
How Does a Radical Equation Work?
A radical equation is similar to any other type of equation, except that the variable is under the radical sign. The difference between the radical equation 3 + √y = 6 and the equation 3 + x = 6 is an additional step, but the principle in solving both equations is the same. If √y = 6 – 3, then √y ) = 3. Similarly, if x = 6- 3, then x = 3.
What Is the First Step in Solving a Radical Equation?
The first step in solving a radical equation is similar to the first step in solving any equation. Isolate the variable on one side of the equation by performing the operations needed. Suppose the equation were √y +4 =12. In order to isolate the variable on the left side √y + 4 – 4 = 12 – 4, or √y = 9. Similarly, if the equation were 2√x = 5, in order to isolate the variable on the left side, (2√x)/2 = 5/2.
What Happens Next?
After the variable has been isolated on the left side of the equation, then the radical can be solved. Since according to the definition (√x)2 = x for any positive number or 0, the radical can be eliminated by squaring both sides of the equation. Therefore, if √y = 3, then y must equal 9, because the square root of 9 equals 3. Also, if √x = 5/2, then x = (5/2)2 or 25/4.
When Is a Solution not a Solution?
It is always important to check the solution in the original equation, because sometimes the “solution” doesn’t fit. Suppose the equation is 5 + √x = 3. According to the first step, √x = 3 – 5 or √x = -2. If both sides were squared, (√x)2 =(-2)2 or x would equal 4. However, 5 +√4 equals 7 and 7 does not equal 3. There is not a way that x could equal both 7 and 3, so that problem has no solution. (Some textbooks call it an apparent solution, and others call it an extraneous solution.)
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