 In this video we provide the solution to question number 11 for practice exam number 4 for math 1050 And we have this exponential equation 2 times 49 to the x plus 11 times 7 to the x plus 5 is equal to 0 We want to solve this exponential equation and find all real solutions Okay So the first thing to notice here is that you have this 7x and a 49x the relationship between 49 and 7 of course is that 7 squared is 49 what that tells us is if I take 7 to the x and I square it That is going to give you 49 to the x by usual exponent laws And so if we insert a new symbol, let's say u is equals to 7 to the x here Then you squared would equal 49 to the x and this equation then can be rewritten as 2 u squared Plus 11 u plus 5 is equal to 0 and therefore this Equation has a quadratic form. Let's first solve the quadratic equation and then we'll switch it back to an exponential in just a second This quadratic equation we could solve by the methods. We know we could factor it. We could Complete the square because the quadratic formula it doesn't matter which one you use I noticed that 2 times 5 gives me 10 2 and 5 gives you 10 for which 10 can be factored as 10 times 1. We also have 10 plus 1 is equal to 11 Right, so there is a magic pair. I can factor this by grouping. So I'm gonna break apart 11 11 11 u here It's gonna become 10 u and just you like so 1 u and so you get 2 u squared plus 10 u That'll be my first group and then for my second group. We're gonna have u plus 5 In the first group, we can take out a 2 u that leaves behind u plus 5 and the second group The only thing we can take out is a 1 but that leaves behind u plus 5 But that's actually great because that's what I wanted But after all since we had a magic pair, it's gonna have to factor that way We factor out the u plus 5 we then end up with 2 u plus 1 times u plus 5 Equals 0 we're then going to set both of these Factors equal to 0 so we get 2 u plus 1 equals 0 We also get u plus 5 equals 0 in the first case the 2 u plus 1 equals 0 minus 1 divide by 2 We see you equals negative 1 half in the second case u plus 5 equals 0 So track 5 from both sides we get negative 5 now that solves the quadratic equation in terms of the symbol u We have to solve it in terms of x right so substitute back in the 7 to the x there So we have 7 to the x is equal to negative 1 half and we have 7 to the x is equal to negative 5 Now remember when it comes to an exponential function 7 to the x is always positive it can never equal a negative so 7 to the x can't equal a negative But also if you went forward with this problem not recognizing that you would take the log base 7 of say like negative 5 And you're gonna find out oh that doesn't exist all right. I should say that's not a real number It's an imaginary number so in particular since the instructions did of course tell us real solutions There's no way that an exponential can equal a negative for a real variable there So we actually have to report that on this problem. There's no solution because there's no real number x That'll satisfy this and that does happen from time to time