![]() Real numbers, you can't take the square root. Less than 0, this is going to be a negative number right hereĪnd you're going to have to take the square root And then what happens ifī squared minus 4ac is less than 0? Well if b squared minus 4ac is Is equal to 0, you only have one solution. Plus or minus, it's not going to be relevant. So in that situation, theĪctual solution of the equation is going to be You add or subtract 0, you're going to get the same value. So it's going to be negativeī plus or minus 0. Radical is equal to 0, you're just going to have the Minus 4ac is equal to 0? If this expression under the When you subtract it from negative b you are going Negative b you're going to get one value for the numerator, and Is greater than 0, what's going to happen? Well, then it's aĪ square root. ![]() If this b squared minus 4ac isĪ positive number- so let's think about this a little bit. Now the reason why this can beĢ solutions is that we have a plus or minus here. Minus the square root of b squared minus 4ac. The solution if it exists is going to be- negative b plus or Plus c is equal to 0, that the solutions are going to be- or Us that if we have an equation of the form ax squared plus bx But what I want to do here isĪctually explore the quadratic formula, and think about how weĬan determine the number of solutions without even maybe We could factor it and justįigure out the values of x that satisfy it and Solutions to the quadratic equation, x squared plus 14x I'm sure there are better real world examples! I hope this helps some, though. Because a parabola and a line can intersect in two places, you might get two answers, and both might be correct. You might be asked where their trajectories overlap (not necessarily at the same time). However, you might also have a problem in which there are two objects moving (maybe, two vehicles of some kind), and one is moving in a straight line and one is moving in a parabola. Often in that case you might get one solution with positive time, and one solution at a negative value of time, so that one you'd throw out. If you are asked to calculate when it hits the ground, you will get two solutions, but both of those can't be right, because physically that makes no sense. For example, if you throw a ball in the air, its height over time can be described as a parabola (quadratic equation). In the real physical world, there are cases in which both these solutions will be valid, and cases where they won't, so you need to pay attention to what the solutions actually mean. So if you have an equation like x^2 + 5x + 6 = 0, it can have two solutions. A parabola, though, curves, so it can cross the x axis in two places. A line can only cross another line in at most one place, so there can be at most one solution (x=3). Thinking of it graphically, if you are solving a linear equation, such as x-3 = 0, this is equivalent to looking at a line y = x-3 and seeing where it crosses the x axis (where y=0). From the mathematical standpoint, two or more answers just means that there is more than one value for the variable that will solve the equation.
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