Solve each equation in Exercises 83–108 by the method of your cho...

Question

Solve each equation in Exercises 83–108 by the method of your choice.

x^2 - 4x + 29 = 0

Accepted Answer

Hello today we're gonna be solving the following equation using any of our preferred methods. So the equation given to us is nine x squared minus 30. X plus 50 is equal to zero. Now we can go ahead and solve this using the quadratic formula. The quadratic formula is when X is equal to negative B plus or minus the square root of B squared minus four times a times C. All of that over two times A. Now a B and C are just going to be the leading coefficients in our given trin Amiel. So what the leading coefficients are going to be 9 - and 50. So a is equal to nine. B is equal to negative 30 and C is equal to 50. Now we just need to go ahead and plug this into our quadratic formula. So X is equal to negative B which is negative negative 30 so positive 30 plus or minus the square root of negative 30 squared minus four times A which is nine times C which is 50. All of that over two times a which is two times 9. Now we need to go ahead and simplify negative 30 squared is going to give us the value of 900 negative four times nine times 50 is going to give us negative 1800 so X is equal to 30 plus or minus the square root of 900 minus 1800. All over two times 9 which is 18. Now 900 -1800 is going to give us -900. So we have X is equal to 30 plus or minus. The square root of negative 900. All over 18. Now the problem here is that we have a negative number inside the square root but we have a way to represent negative numbers inside the square root. Suppose you have the square root of negative X. You can rewrite this as the square root of X times I because whenever you have a negative number inside the square root it produces an imaginary number I So we can go ahead and rewrite the square root of negative 900 as the square root of 900. I So X is equal to 30 plus or minus the square root of 900 I over 18. Now the square root of 900 can be evaluated to so X is equal to 30 plus or minus I all over 18 And in the numerator we can go ahead and factor out a 30. So doing so is going to give me X is equal to 30 times one plus or minus I all over 18 And 30/18 can reduce to 5/3. So X is equal five times one plus or minus I all over three. Now this is the most simplified version of our answer but we need to go ahead and write this in standard form because we're using an imaginary number. So in order to do that we're going to go ahead and redistribute five back into the quantity one plus or minus I. And that's going to give us X is equal to five plus or minus five I over three. And now we can go ahead and separate this into two fractions because X is going to equal to 5/3 plus or minus 5/3. I and now finally we can just go ahead and separate the values of X and X is going to equal to 5/3 plus 5/3. I and X is equal to 5/ minus 5/3 I. And these are gonna be the two solutions to our equation. And since these are the two solutions to our equation, the answer to this problem is going to be C. So I hope this video helps you in understanding how to solve this problem using the quadratic formula and I'll go ahead and see you all in the next video.

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