Would make one of these, Y equals five would make this zero, which would make theĮntire expression zero, so this is a solution. This one equal to zero? Well Y would be equal to negative three. So the solutions are gonna be how do you make this one equal zero? Well Y would be equal to five. Product of two things that are equal to zero that means that you could get this to equal zero by making one or both But we now know how to factor, this is gonna be zero isĮqual to, we can write this as Y minus five times Y plus three. Do a search on Kahn Academy for factoring quadratic expressions. And if what I'm doing right now looks a little bit like voodoo to you, I encourage you to reviewįactoring quadratic expressions on Kahn Academy. Product is negative 15 their sum is negative two. So it's gonna be negative five and three. Well if the sum is negative that means that the negative number has Numbers three and five, either negative three, positive five or positive three, negative five jump out and whose sum is negative two. Would do that is first of all kind of think of two numbers Quadratic right over here, we could factor this quadratic, we could factor this quadratic expression. And now to solve for the Ys that would satisfy this Write it the other way, Y squared, let me write the minus 2y minus 15. Side you're gonna have Y squared minus 15 minus, actually let me And then on the left hand side we're gonna be left with zero. And we could subtract six from both sides. So if 2y plus six is equal to Y squared minus nine. Then can solve for Y, the Y of a solution to this system. Well this one actually canīe solved with substitution because 2y plus six needs to be equal to X but then we also that X isĮqual to Y squared minus nine. The following represents all solutions X comma Y to the system of equations shown below? This is an interesting system of equations because this is a linearĮquation, this first one, but the second one is nonlinear. So the \(n\) th term of the quadratic sequence is \(n^2 + 5n + 3\). The coefficient of \(n^2\) is half the second difference, which is 1. The sequence will contain \(2n^2\), so use this: \ The coefficient of \(n^2\) is half the second difference, which is 2. The second difference is the same so the sequence is quadratic and will contain an \(n^2\) term. Work out the \(n\) th term of the sequence 5, 11, 21, 35. The \(n\) th term of this sequence is therefore \(n^2 + 1\). In this example, you need to add 1 to \(n^2\) to match the sequence. To work out the \(n\) th term of the sequence, write out the numbers in the sequence \(n^2\) and compare this sequence with the sequence in the question. Half of 2 is 1, so the coefficient of \(n^2\) is 1. In this example, the second difference is 2. The coefficient of \(n^2\) is always half of the second difference. The sequence is quadratic and will contain an \(n^2\) term. The first differences are not the same, so work out the second differences. Work out the first differences between the terms. Work out the \(nth\) term of the sequence 2, 5, 10, 17, 26. The first five terms of the sequence: \(n^2 + 3n - 5\) are -1, 5, 13, 23, 35 Finding the nth term of a quadratic Example 1 Write the first five terms of the sequence \(n^2 + 3n - 5\). Terms of a quadratic sequence can be worked out in the same way. The \(n\) th term for a quadratic sequence has a term that contains \(n^2\). They can be identified by the fact that the differences between the terms are not equal, but the second differences between terms are equal. Quadratic sequences are sequences that include an \(n^2\) term. Finding the nth term of quadratic sequences - Higher
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