Solve each equation using completing the square. See Examples 3 a...

Question

Solve each equation using completing the square. See Examples 3 and 4. 3x^2 - 9x + 7 = 0

Accepted Answer

Hello, everyone. We're going to solve the equation. Four X squared minus 24 X plus 111 equals zero by completing the square. First thing I recall about completing the square is that I don't want my equation to have a leading coefficient, meaning I don't want to value in front of my X squared. Since this one has a value of four, I'm gonna divide each term by four. So that way cancels out and it stays balanced. So when I do that, I get X squared minus six X plus 111 4th equals zero. Now, next thing I know I need to do for completing the square, I need to move my constant to the right hand side of the equation so I can create a perfect square, try no meal on the left hand side. So on the left hand side, I'm going to leave my X squared minus six X and I'm subtracting 111 4th from both sides. So I have X squared minus six X equals negative 111 4th. I left a space after the six X because that's where I'm going to put my new term So I create that term, I recall it is the B term divided by two then squared. So negative six, divided by two is negative three, negative three squared is a positive nine. And since I added it on the left hand side, I also need to add it on the right hand side, so it stays balanced. So right now I have X squared minus six X plus nine equals negative 111 4th plus nine. I'm gonna rewrite the left hand side as a binomial square. So it's X -3 squared. On the right hand side, I want to combine my negative 111/4 and nine recall that when you add a fraction, you need to have a common denominator. So nine is actually the same thing in this case as 36/4. So when I combine those, I get negative 75/4. So now I want to solve for X, since I have a square on the X, on the left hand side, I'm gonna take the square root of that whole side. And what I do to one side, I have to do the other. So the left hand side, the square and the square root cancel out. Leaving me with X -3. On the right hand side, I recall that if I am taking the square root of a fraction, I need to do the square to the numerator and the square root of the denominator, I have the square root of -75 over the square root of four. And I need to clean that up a little bit. I want to simplify things there. I don't want to leave that with that many radicals. So off to the side, I'm gonna figure out what is the simplified version of the square root of negative 75? I recall that the square root of a negative one is the same as I. So it's an imaginary number. And then the square root of 75, I can break down to the squared of 25 times the square to three square to 25 is five square to three stays the square to three. So this is plus or minus five I radical three. So I'm gonna go plug that back in, I'm actually gonna put the plus or minus out front of my fraction. So in the numerator, I have five, I times the square to three in the denominator, the square to four is two. So we're almost to the value of X to get X by itself, I need to add three to both sides. And when I do that, I get X equals three plus or minus five I radical 3/2. And for our solution, we want to split that into two X values. So it would be three minus five I Radical 3/2 and it would be three plus five I radical 3/2. So those are two X values and when I check the possible answers that matches to answer choice. D have a nice day.

Solve each equation using completing the square. See Examples 3 and 4. 3x^2 - 9x + 7 = 0

Key Concepts

Completing the Square

Quadratic Equations

Discriminant

Watch next