 So let's try to solve the equation square root x minus one equals x minus three So we can solve this equation by squaring both sides So over on the left-hand side we have the square of a square root Which will eliminate the square root leaving us with just x minus one Over on the right-hand side. We have x minus three squared So we'll expand that and we'll get a new equation x minus one equals x squared minus 6x plus 9 Since this is a quadratic equation. We have an x squared. We'll want to get all of our terms onto one side So over on the left-hand side, we have an x and so we'll get rid of it by subtracting an x We have a minus one, so we'll get rid of it by adding one and we have to do the same thing to both sides So we'll subtract x and add one to the right-hand side as well And that gives us our new equation And since this is a quadratic equation, we can solve it. We'll use the quadratic formula So we know that our equation has Coefficient of x squared equal to one Coefficient of x equal to minus seven and constant coefficient equal to ten So we'll place these values in the quadratic formula and then simplify and that gives us our solutions five or two Now the disadvantage of the quadratic formula is that it always works on any type of quadratic and doesn't require a lot of trial and error But if you want something that doesn't always work Doesn't always apply and involves a lot of extra work. We could try to factor So we want x squared minus seven x plus ten to be the product of two things Our first terms have to multiply out to x squared. So they have to be x and x Our second terms have to multiply out to plus ten So they've got to be a b where a b is equal to ten and we list every possible pair of numbers that multiply to ten And at this point the only thing we can do is to try every one of these until we either find a factorization Or determine that none of them work. So let's try one and ten So we want to see if x squared minus seven x plus ten is equal to the product x plus one times x plus ten So we'll expand simplify and They are not equal So we try our next possibilities minus one and minus ten So we want to see if x squared minus seven x plus ten is equal to x minus one times x minus ten So we expand and find that they are not equal Third times the charm so two and five have to work So we'll check it out though x plus two times x plus five is going to be unequal Since the universe is a kind and gentle place We know that negative two and negative five have to be the solution. So we won't even bother to check them I would let's check it out Do we actually get a factorization x minus two times x minus five will expand and we find that we are in fact successful But we're not done yet. All we did was factor x squared minus seven x plus ten We have not yet solved the equation. So we have to go through another step Our equation is x squared minus seven x plus ten and all the work that we just did Gave us the factorization for the left-hand side. We have product equal to zero So we know that one of the factors is zero either x minus two is zero and we can solve giving us our solution x equals two or The other factor x minus five is equal to zero so we can solve Giving us the other solution x equals five and it doesn't really matter how we solve it We get the same answer either way We can either use the quadratic equation, which will give us our solution immediately and for all possible quadratics Or we could go through trial and error factorization And plus this additional step to possibly solve it as long as our quadratic is factorable In a kind and gentle universe, you'd always get factorable quadratics Unfortunately, we don't live in that universe You're probably better off using the quadratic formula What important thing to remember is that because we squared a square root It's possible we introduce some extraneous solutions and in general you should always check your solutions But this is especially true if the equations are more complicated than a simple linear equation really However, we solve the equation. We must check to see if the solutions are extraneous so if x equals two will Substitute this into our original equation and see if we truly get a true statement We'll simplify And at this point it's useful to remember that the principle square root of n is never negative So we know that this statement principle square root of one possibly equal to negative one is false So x equals two is not a solution We can say that x equals two is an extraneous solution The other possibility we have is x equals five. So if x is equal to five will substitute Will simplify and principle square root of four is two So x equals five does give us a true statement. So x equals five is a solution And if it's not written down it didn't happen. We might want to say the only solution is x equals five