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How Do You Convert a Quadratic from Standard Form to Vertex Form by Completing the Square if a≠1?

Write y = 2x2 + 12x – 4 in vertex form by completing the square.

Summary

  1. Our equation is in standard form to begin with: y=ax2+bx+c
  2. We want to put it into vertex form: y=a(x-h)2+k
  3. We can convert to vertex form by completing the square on the right hand side
  4. Adding 18 to both sides gives us a perfect square trinomial on the right
  5. The vertex of a quadratic equation in vertex form is (h,k), so our vertex is (-3,-22)

Notes

    1. Right now our quadratic equation, y=2x2+12x-4 is in standard form
    2. We want to get it into vertex form
    3. To do this, we are going to use the method of completing the square
    1. Our equation is y=2x2+12x-4
    2. So a = 2, b = 12, and c = -4 in our equation
    3. Vertex form allows us to easily pick out the vertex of a quadratic function
    4. Remember, the vertex is the point where a parabola crosses its axis of symmetry
    1. Usually when we are completing the square, we are trying to solve for 'x'
    2. Here we're not actually trying to find a value for 'x' - we're just using the same method to rearrange the equation
    3. Since vertex form also has an 'x' and a 'y', we need to keep that 'y' in our equation while we complete the square on the other side
    1. A perfect square trinomial is a trinomial that can be factored into something squared
    2. So when you factor it, you would have two identical binomials being multiplied together
    1. We want to be able to factor the right hand side into two identical binomials
    2. We can't do that if our last term is -4, so we need to move it over to the other side
    3. Then we can figure out what we DO need to add to get it to factor how we want it
    1. Make sure to add 4 on BOTH sides to preserve the equality!
    2. The 4's cancel out on the right, leaving us with just 2x2+12x
    1. In order to complete the square, we want the coefficient in front of x2 to be 1
    2. So we need to factor a 2 out of the right hand side first
    3. Then we can complete the square with what is left
    4. We can just ignore the 2 for now and focus on the x2+6x
    5. Don't forget about the 2 though - we'll come back to it later!
    1. A perfect square trinomial has the form ax2+bx+c, just like standard form
    2. So 'c' is the constant term we're adding at the end
    3. We want to find a value for 'c' that will allow us to factor the right hand side into two identical binomials
    1. A perfect square trinomial factors into the product of two identical binomials
    2. Since our x2 term doesn't have a coefficient in front of it, we know the first terms of our binomials will be 'x'
    3. Since the middle term of the trinomial is positive, the signs of our binomials will be positive as well
    4. When we multiply out (x+3) times (x+3), we will get 3x+3x for our middle terms, which adds up to 6x
    5. For our last term we would get 3•3, or 9, which is our value for 'c'
    1. 9 is the number that we need to add INSIDE the parentheses
    2. But we're still multiplying everything in the parentheses by 2
    3. So we can't just add 9 to both sides - we need to add 9 inside the parentheses then multiply the 2 back through
    4. Then we can find out what number we ACTUALLY need to add to the left hand side to keep the equation equal
    1. We want to add 9 inside the parentheses to create the perfect square trinomial we were trying to make
    2. When we distribute the 2, we can see that we are actually adding 18 to the right hand side
    1. When we multiplied out the right hand side, we saw that we were actually adding 18
    2. So to keep the equation equal, we need to add 18 to the left as well
    1. We already had the right hand side factored from before, so this should be pretty easy!
    1. Factoring out the 2 gives us the perfect square trinomial we had in the parentheses before
    1. Perfect square trinomials factor into the product of two identical binomials
    2. And we've done that already too!
    3. We used the factored form to figure out how to make the perfect square trinomial in the first place!
    4. Multiplying two identical things is the same as squaring that thing, so we can rewrite (x+3)•(x+3) as (x+3)2
    1. Our equation is ALMOST in vertex form - but we need to have 'y' by itself on one side
    2. In order to do that, we need to subtract 22 from both sides to get rid of it on the left
    1. Make sure to subtract 22 from BOTH sides to preserve the equality!
    2. The 22's cancel on the left, leaving 'y' by itself, which is what we wanted!
    1. Remember that vertex form of a quadratic equation is y=a(x-h)2+k
    2. Then the vertex is the point (h,k)
    3. Notice that in vertex form, the 'h' has a minus sign in front of it
    4. This means that when we pick out the value for 'h', we need to flip the sign that's in front of it
    5. So since the 3 has a plus sign in front of it, our value for 'h' is -3
    6. But 'k' has a plus sign in front of it in vertex form, so the sign stays the same and our value for 'k' is -22