Completing the square
Completing the square
is a useful tool made for the conversion of a quadratic equation in standard form
(y = ax2+bx+c) to vertex form (y = a(x-p)2 +q). It’s
helpful because after completing the square, when the equation is in vertex
form, you gain access to beneficial information that makes equations easier to
solve. This includes the direction of opening, vertex, axis of symmetry,
maximum/minimum value, range, and domain of a parabola.
There are 7 simple
steps to completing the square:
Original equation –> y = x^2 - 8x -1
Original equation –> y = x^2 - 8x -1
- Group first two terms. y = (x^2-8x) -1
- If the a doesn’t equal 1, factor it out of the brackets and
place it in front. (in this case it does) y = 1(x^2-8x)
-1
- To create a perfect square trinomial take the b value, divide it by two, and then take the square
of the quotient. -8/2 = -4^2 = 16
- Add and subtract this value inside the brackets to
keep balance. y = 1 (x^2-8x+16-16) -1
- Pull out the negative number from inside the brackets. Make sure to
multiply it by the a value that is in front of the brackets. Place it to
the right of the brackets. y = 1 (x^2-8x+16) -16 -1
- Now combine the terms. y = 1(x^2-8x+16) -17
- You have now created a perfect square trinomial and it can be written in
factored form.
y = 1(x-4)^2– 17
From here you can solve the equation, if needed, and find the value of x by making y = 0. The video below provides more of a visual aid on how to find the value of x along with completing the square.