![]() And just like we saw before, that means that x minus two is equal to the positive or negative square root of nine. Sides and so we could get x minus two squared is equal to nine. So we could rewrite this as x, x minus two squared minus nine equals zero. This thing right over here "equal zero?" So, let me just write that down. X-values where the graph of y equals f of x intersects the x-axis, this is equivalent to saying, "For what x-values doesį of x equal zero?" So we could just say, "For what x-values does And that means that ourįunction is equal to zero. Notice our y-coordinateĪt either of those points are going to be equal to zero. Well, in order to intersect the x-axis, y must be equal to zero. The x-values where you intersect, where you intersect the x-axis. Let's say that the y isĮqual to some other function, not necessarily this f of x. If I have the graph of some function that looks something like that. Talking about some graph, so I'm not necessarily gonnaĭraw that y equals f of x. And then we're asked at what x-values does the graph of y equalsį of x intersect the x-axis. So, we are told that f of x isĮqual to x minus two squared minus nine. That's presented to us in a slightly different way. So, these are the two possible x-values that satisfy the equation. And when x is equal to negative five, negative five plus three is negative two, squared is positive four, minusįour is also equal to zero. You substitute it back in if you substitute x equals negative one, then x plus three is equal to two, two-squared is four, minus four is zero. Substitute it back in, and then you can see when So, those are the two possible solutions and you can verify that. Negative two minus three is negative five. Or, over here we could subtract three from both sides to solve for x. Sides to solve for x and we're left with x isĮqual to negative one. So, if x plus three isĮqual to two, we could just subtract three from both If x plus three was negative two, negative two-squared is equal to four. Notice, if x plus three was positive two, two-squared is equal to four. And so we could write that x plus three couldīe equal to positive two or x plus three could beĮqual to negative two. Positive square root of four or the negative square root of four. Something right over here, is going to be equal to the If something-squared is equal to four, that means that the something, that means that this Three is going to be equal to the plus or minus So, one way of thinking about it is, I'm saying that x plus Way of thinking about it, if I have something-squared equaling four, I could say that that something needs to either be positive or negative two. And so now, I could take the square root of both sides and, or, another So, x plus three squared is equal to four. ![]() So, adding four to both sides will get rid of thisįour, subtracting four, this negative four on the left-hand side. This is I'm gonna isolate the x plus three squared on one side and the best way to do that But this solution does find useful applications in the further analysis of the convergence problem for continued fractions with complex elements.The video and see if you can solve for x here. This general solution of monic quadratic equations with complex coefficients is usually not very useful for obtaining rational approximations to the roots, because the criteria are circular (that is, the relative magnitudes of the two roots must be known before we can conclude that the fraction converges, in most cases). In case 2, the rate of convergence depends on the absolute value of the ratio between the two roots: the farther that ratio is from unity, the more quickly the continued fraction converges. ![]() If the discriminant is not zero, and | r 1| = | r 2|, the continued fraction diverges by oscillation. ![]() If the discriminant is not zero, and | r 1| ≠ | r 2|, the continued fraction converges to the root of maximum modulus (i.e., to the root with the greater absolute value).If the discriminant is zero the fraction converges to the single root of multiplicity two.The general form isĪ x 2 + b x + c = 0, Ĭonverges or not depending on the value of the discriminant, b 2 − 4 c, and on the relative magnitude of its two roots.ĭenoting the two roots by r 1 and r 2 we distinguish three cases. In mathematics, a quadratic equation is a polynomial equation of the second degree. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |