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Solving Radical Equations - Problem 4
Okay I want to go through in this situation and solve for x. What I’m going to need to is get that square root all by itself, get it isolated and then I’m going to square both sides of this equation. So this square root is not isolated, it has this -x attached to it.
I‘m going to add x to both sides so that now my square root will be isolated, oops it’s not a parenthesis. X squared minus 4x minus 1. Okay if I want to take the square root of a trinomial, the first thing I’m going to think about is whether or not that trinomial is a perfect square trinomial. Like sometimes I might be able to recognize that this would be a binomial times itself and then taking the square root would be really easy. This situation it’s not, bummer.
So what I need to do is square both sides of this equation. Be really careful this is what a bonehead is going to write, don’t be the bonehead. Some bonehead is going to write 9 plus x² is equal to x² minus 4x minus 1. What this person did is they squared both sides right? No, come on you guys know better than that. When you square a binomial, you have to FOIL, don’t forget to FOIL. This person forgot the inners and outers. This side is correct though.
Okay let’s revisit the Foiling. FOIL; first is 9, outers is +3x, inners is +3x and then the lasts is the x², so the correct answer we’ll leave the bonehead behind will look like this, 9 plus 6x plus x² is equal to x² minus 4x minus 1. I kind of misspoke this isn’t the final answer; this is just what happens the result after I square both sides.
Okay so now you just solve for x and it’s kind of scary because I have x² terms on both sides of the equals, but fortunately if I subtract x² from both sides those would be eliminated. I have 9 plus 6x is equal to -4x minus 1. I want to combine all my Xs, so I would add 4x to both sides, subtract 9 from both sides and I would have 10x equals -10 divide both sides by 10 and I get x equals -1.
All right I think that’s my answer, but I did a whole lot of Mathy stuff, so I’m just going to check and make sure I did it right. Let me show you how to check. You have your x value that gets substituted up into here and hope your answer is equal to 3. Let’s try it.
To check I’m hoping that 3 is equal to -1 or the opposite of -1 is +1, plus the square root of -1² is +1, -4 times -1 is +4 and then take away 1. So I’m hoping that 3 is equal to 1 plus the square root of 4 which it is, 3 is equal to 1 plus 2, sweet, that means I did it.
So you guys whenever you see an equation that has a radical or a square root, your job is to get that square root all by itself and then multiply or excuse me to square both sides, but please, please, please don’t be the bonehead. Make sure if you have a binomial make sure you’re Foiling. That will drive your teacher nuts if you don’t do it correctly.