Solve each equation. (2x+1)(x-4) = x

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Accepted Answer

Hello, today we're going to be solving the following quadratic equation. So what we are given is three X minus four times X minus three is equal to five X. Now, in order to start solving for X, we're gonna go ahead and foil the left hand side together. So in order to do that, we're going to multiply three X with X which is going to give us three X squared, then we're going to multiply three X with negative three, which is going to give us negative nine X. Then we're gonna multiply negative four with X which is going to give us negative four X. And finally, we're going to multiply negative four with negative three, which is going to give us positive 12 and that is still equal to five X. On the left hand side, we're going to combine the like terms together. And that means we're going to combine negative nine X with negative four X and that is going to give us negative 13 X. So we have three X squared minus 13, X plus 12 is equal to five X. And since we want to solve for X, we want to set this equation equal to zero. So we're going to subtract five X to both sides of the equation. And what we are left with is three X squared minus 18 X plus 12 is equal to zero. Now, one thing to notice here is that the coefficients of each term are all multiples of three. So we can go ahead and divide every term by three. And that is going to give us X squared minus six, X Plus four is equal to zero. Now, in order to solve this quadratic equation, in order to solve this equation, we're going to use the quadratic equation. And the quadratic equation states that X is equal to negative B plus or minus the square root of B squared minus four times A times C all over two times A. And this is where A B and C are going to be the coefficients of the given Trin Amiel. So in our equation, the coefficient of the first term is going to be one. So A is going to equal to one. The coefficient of the second term is negative six. So B is equal to negative six and the constant is going to be positive four. So C is equal to four. So if we take these three values and plug it into the quadratic equation, what we have is X is equal to negative negative six, which is going to give us positive six plus or minus the square root of negative six squared minus four times A which is one times C which is four all over two times A which is two times one. And now we just need to simplify in the numerator negative six squared is going to give us positive 36 negative four times one times four is going to give us negative 16. So we have six plus or minus the square root of 36 minus 16, all divided by two times one, which is two And 36 -16 is going to give us 20. So what we have is six plus or minus the square root of all divided by two. Now 20 is not a perfect square, but we can rewrite 20 as a product of prime numbers C 20 can be broken up as 10 times two and 10 can be broken up as five times two. And this is where both twos and the five, our prime numbers. So we can go ahead and rewrite the square root of 20 as the square root of two times two, which is two squared times five. And the square root is going to allow us to take out two squared as a single two in front of the square root. So to summarize the square root of 20 can be written as six plus or minus two times the square root of five all divided by two. Now, in the numerator, each term has a coefficient that is a multiple of two. So we can factor out a two from the numerator. And that is going to give us two times three plus or minus the square root of five divided by two. And since we have a two in the numerator and denominator, those are going to cancel each other out. And that is going to leave us with X is equal to three plus or minus the square root of five. And since this is plus or minus statement, we can go ahead and separate this into two equations that is going to give us X is equal to three plus the square root of five and X is equal to three minus the square root of five. And these are going to be the two solutions for X for the given quadratic equation. And with that being said, the answer to this problem is going to be D. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Solve each equation. (2x+1)(x-4) = x

Key Concepts

Factoring

Distributive Property

Quadratic Equations

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