Squares and Square Roots
A radical equation is an equation that features a variable contained inside a radicand. At least it won't get wet if it rains.
An example of a radical equation is .
The equation is not a radical equation, because the variable doesn't occur inside the radicand. The 5 and 9 are making it wait outside, apparently, and it's getting drenched.
To solve a radical equation that has a single radical on one side and a number on the other, we square both sides to eliminate the radical and then solve the resulting equation. It will only become more complicated from here, so enjoy it while it lasts.
We square both sides to eliminate the radical. This gives us x + 1 = 49, so x = 48.
Check to make sure this works. Plugging x = 48 into the left-hand side of the equation yields , which is indeed 7.
Sometimes we need to rearrange the equation a little, so that the radical is all by itself, or "isolated" on one side of the equation. Poor little radical.
First we need to add 1 to each side to get the radical all by itself on the left-hand side of the equation. Don't feel badly for him; he's a loner anyway. He'll appreciate the solitude:
Then, we square both sides to eliminate the radical:
5 + x = 16.
Solving this equation gives us
x = 11.
We'll push our double-checking agenda again here. Plugging x = 11 into the left-hand side of the equation yields
which is, in fact, 3. Looks like our train hasn't come off the tracks. The two sides of the equation agree, so our solution is
x = 11.
As we mentioned, some radical equations have no solution, and are so obvious about it that we don't need to do any work. We have no problem with that. Subtlety is overrated.
The radical is already by itself. It even hung a "Do Not Disturb" sign on its door. We square both sides:
2x + 6 = x2 + 6x + 9.
Now we rearrange for a nice quadratic equation, being sure we've got zero off to one side:
0 = x2 + 4x + 3.
This equation factors as
0 = (x + 3)(x + 1),
so our solutions are
x = -3, x = -1.
As always, we'll cover our butts by checking to make sure both of these are correct solutions.
When x = -3, the left-hand side of the original equation is
and the right-hand side of the original equation is
-3 + 3 = 0.
Since the two sides agree, x = -3 is a solution to the original equation.
When x = -1, the left-hand side of the original equation is
and the right-hand side is
(-1) + 3 = 2,
so x = -1 is also a solution to the original equation. Excellent. Butts officially covered.
Be careful: When solving radical equations, make sure to check your answers. Sometimes they won't actually be solutions to the original equation. You don't want egg on your face. Especially if it's been soft-boiled.
We square both sides to find
4x + 5 = x2
and rearrange for the nice quadratic equation
0 = x2 – 4x – 5.
This factors as
0 = (x – 5)(x + 1)
x = 5, x = -1.
Now we check to make sure both of these are solutions to the original equation.
When x = 5, the left-hand side of the original equation is , and the right-hand side of the original equation is also 5, so x = 5 is a solution.
When x = -1, the left-hand side of the original equation is .
However, when x = -1 the right-hand side of the original equation is (-1).
Since the left-hand and right-hand sides of the equation do not agree for x = -1, this isn't a solution. Argh, so close. x = -1 is an extraneous solution. In other words, it's a number we found through legitimate means, but it's not a solution to the original equation.
It's like if you put in an honest day's work and then got paid in Trident Layers. You got something for your efforts, but it's not what you wanted, and it won't help you pay for anything. Although at least it does taste exactly like green apple and golden pineapple. Well, that's something.
Our final answer for this problem is the single solution x = 5.