Information described below to the designated agent listed below. Or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one We now set each factor equal to zero and solve for x. Thus, when factored, the original equation becomes (2 x + 1)( x – 4) = 0. We will next factor out a 2 x from the first two terms. We will then group the first two terms and the last two terms. Those two numbers which multiply to give –8 and add to give –7 are –8 and 1. We must now think of two numbers that will multiply to give us –8, but will add to give us –7 (the coefficient in front of the x term). (We could use the quadratic formula, but it's easier to factor when we can.)īecause the coefficient in front of the is not equal to 1, we need to multiply this coefficient by the constant, which is –4. In other words, to find the roots of a function, we must set the function equal to zero and solve for the possible values of x. Whenever a function passes through a point on the x-axis, the value of the function is zero. The roots of a function are the x intercepts of the function. Therefore, because 9x 2 – 6x + 4 is not a perfect square, it doesn't have exactly one root. It might be tempting to think that 9x 2 - 6x + 4 = (3x - 2) 2, but it does NOT, because (3x – 2) 2 = 9x 2 – 12x + 4. This CANNOT be written as a perfect square, because it is not in the form a 2x 2 + 2abx + b 2 = (ax + b) 2. This function is also a perfect square and has a single root. We can multiply both sides by four to get rid of the fraction. If we look at f(x) = x 2 –2x + 1, we see that x 2 – 2x + 1 is also a perfect square, because it could be written as (x – 1) 2. Thus, this equation has only one root, and it can't be the answer. We notice that 4x 2 - 4x + 1 is a perfect square, since we could write it as (2x – 1) 2. Let's examine the choice f(x) = 4x 2 – 4x+1. Additionally, a quadratic equation is a perfect square if it can be written in the form a 2x 2 + 2abx + b 2 = (ax + b) 2. If we set (x – a) 2 = 0 in order to find the root, we see that a is the only value that solves the equation, and thus a is the only root. This is because a quadratic function that is a perfect square can be written in the form (x – a) 2. If a quadratic function has one root, then it must be a perfect square. We can set each function equal to zero and determine which functions have one root, and which does not.Īnother piece of information will help. The roots of an equation are the points at which the function equals zero.
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