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 + 2x = 5

Accepted Answer

Welcome back. I am so glad you're here. We are given this equation 28 X squared plus nine X equals 37. We are asked to complete the square and then solve it. All right, so the first step here in completing the square is we're going to see what format our equation is in and this is in a little bit of a modified quadratic equation. We've got R A X squared plus R B X. And then that's equal to the negative C instead of plus C equals zero. So this is a great starting place for completing the square. The next thing we want to do is see if are a term is equal to one. The A term is in front of the X squared in here is equal to 28. We need that to be equal to one. So we're going to divide everything by 28. So 28 X squared divided by 28 gives us one X squared or just X squared nine X divided by 28 gives us A plus 9/28 X. And then 37 divided by 28 is 37. 28th. Okay, we've got our A term equal to one and next we are going to set it up to complete the square so we are going to take our B term and here the B term is in front of the exits nine 28th 9/28 And then we're going to divide our B term by two. So that whole thing over to and then we are going to square it. So we've got a little bit of work here to simplify that. Once we've done all this work, we're going to add that value to both sides of the equation. So let's simplify. 9 28 divided by two. Can be rewritten like this 9 28 divided by two, which is the same thing as nine 28th times 1/2 and nine times one in the numerator is nine and 28 times two in the denominator is 56. So we've got nine 56th squared now nine squared in the numerator is 81 56 squared in the denominator equals 3136. Alright, so this is going to get added to either side of our equation. So I'll leave a space here, X squared plus nine 28th X. Leave a space equals 37 28th. Now going back in and adding plus 3136 to both sides of the equation. Next we are going to factor the left hand side and factoring it. We've basically set it up for factoring because we know that to get our X square term is going to be X times X. And then this second term here, that is going to be our B over to that we have in the parentheses before it was squared. So RB over to will be plus 56 And if we went and we foiled this whole thing out that would give us our X squared plus 9 28 X plus 81. Over 3136 by design. So there's our completing the square. Now let's solve for X. We need to add these fractions on the right hand side of the equation. They need to have a common denominator and the common denominator is going to be 3136. We need to multiply 37/28 by 112/112. So 37 times 112 gives us 4144. And then in the denominator we have 3, added to 81 over 3,136. Now we can do that addition on the right hand side adding our enumerators. We will have 4225 over our 3136. We can bring down this X plus 9 squared. Alright. To get X by itself. We need to get rid of this squared so we will take the square root of both sides. On the left will have X plus 56. That will be equal to will have plus or minus on the right hand side because we take the square root and the square root of the numerator, 4225 equals 65. The square of the denominator, 3136 is 56. Look at those nice normal numbers And now We can subtract 9/56 from both sides and we will have X equals I'll put the negative nine 56th first plus or minus 65/56. Now we have a common denominator so we can combine these and I'll split these up into two different answers because we have X equals negative 56 plus 65 56 months. And we have X equals negative 9 50 c sixths minus 65 56. And doing the math negative 9 56 plus 66 gives us 56 which is one. So one of our answers is a positive one. And doing the math over here negative 9 56 minus 56 equals a negative 56th. Which simplifies to a negative 28th and there are two answers. One and negative 37 28th. We look at our answer choices and this matches with answer choice, a well done. We'll catch you on the next one

Solve each equation using completing the square. See Examples 3 and 4. 3x^2 + 2x = 5

Key Concepts

Completing the Square

Quadratic Equations

Standard Form of a Quadratic

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