Converting Standard Form To Vertex Form

Question

y = 3x² +15x +29

Step 1 y-29 = 3 (x²+5)

Step 2 y-29 = 3 (x² + 5/2^²)

Step 3 y-29 = 3 ( x²+5x +25/4)

Step 4 y-29 +25/4 = 3( x²+5/2x)

Step 5 y= 3( x+5/2)² +91/4 This is wrong, I am aware. It should end with 41/4.

What step did I error?

Answer 1

If you compare the right hand sides of steps 1 and 3, you'll see that you added 3·(25/4) to the right hand side of the equation in step 3. You must add the same amount to the left hand side of the equation to keep the two sides equal:

y-29+(75/4) = 3 ( x²+5x +25/4)

As written, step 2 is incorrect as well; it should be (x + 5/2)^2. It's actually here that you add the 75/4, you just didn't write it out until step 3. So the equal sign in step 2 is incorrect as written. To avoid this, I'd recommend that you switch steps 3 and 2; that is, do step 3 before step 2 and remember that whatever you add to the RHS you must also add to the LHS to maintain the equality.