rajak01 wrote:
Another follow up question guys, does a quadratic equation always equal a zero although it is not mentioned because one of the GMAT questions assumed that the equation is zero although it is not stated.
See the question here :-
https://gmatclub.com/forum/if-t-8-is-a- ... 66573.htmlIt's important, when learning algebra, to distinguish carefully between an "expression" and an "equation". An expression is just an arithmetic combination of letters and numbers. So 29, 7x, 3x + 11, x^5 - 0.5 and x/9 are five different simple examples of expressions. If you see an expression alone, you have no idea what it is equal to. The only thing you're legally allowed to do with an expression is to rewrite it in ways that do not change its value. So if you saw the expression 3x/6, it would be fine to rewrite that as x/2, since by cancelling the '3' we are not changing the value of the expression. But that's the only useful thing we can do with "3x/6".
An equation tells you that one expression has the same value as another expression. When you have an equation, you can do a ton of things - you can rewrite equations in all kinds of ways, provided you're doing the same thing to both sides (with some minor caveats).
A "quadratic" is a type of expression, one that contains a letter that is squared, often added to a letter to the first power and to a number. The letter terms might be multiplied by a number too. So x^2, or x^2 + 5x + 6 or x^2 - 16 are simple examples of quadratics. If you only see a quadratic on its own, then you're only looking at an expression. You have no idea what it's equal to, so you can't write down any kind of equation - that would always be a serious mathematical mistake. Do not assume that a quadratic expression is equal to zero. You can, however, rewrite a quadratic in equivalent ways, especially by factoring (that's almost always going to be useful). So if you saw the quadratic x^2 - 16, a difference of squares, you could rewrite that as (x + 4)(x - 4).
A "quadratic equation" is an equation that is written, or that can be rewritten, so that it has a quadratic expression on one side and zero on the other. So x^2 = 16, or x^2 - 16 = 0, or x^2 + 5x + 6 = 0 are examples of quadratic equations. When we know that our quadratic is equal to zero, then we can (at least for examples as simple as those on the GMAT) use the familiar quadratic factoring technique to find the solution or solutions for x.
In the question you link to above, the quadratic is not equal to zero. We know that t^2 - kt - 48 has (t - 8) as a factor. The quadratic must have a second factor, which will look like (t + a), where a is some number. So
t^2 - kt - 48 = (t - 8)(t + a)
If you multiply out the right side, you get t^2 + (a - 8)t - 8a. Here (a - 8) and 8a are just numbers, and number at the end on the left side, "-48", must equal the number on the end on the right side, "-8a". So a = 6. We want to find k, and since -k is in front of the "t" on the left side, that must equal (a-8), the number in front of the "t" on the right side. Since a = 6, we know -k = a - 8 = 6 - 8 = -2, and if -k = -2, k = 2.
That's a bit complicated, and some solutions in the thread you link to make use of an important fact. If (t - 8) is a factor of a certain quadratic *expression*, that always means that *if* you set that expression equal to zero, then t = 8 will be a solution of the equation you get. So the quadratic is not inherently equal to zero; instead we're saying "if the quadratic were equal to zero, then 8 would be a solution for t" and drawing conclusions about k from that.