What this lesson covers
- Take half of b, square it, then add and subtract that number to stay balanced
- Factor the perfect square, and the leftover constant gives k
- If a is not 1, factor it out of the x terms first, and remember the pulled-out constant gets multiplied by a
- Vertex form y = a(x - h)squared + k has vertex (h, k); the sign inside flips
Chapters
- 0:00The vertex is the whole game
- 0:22Why it is called completing the square
- 1:02Worked example: x squared minus 6x plus 5
- 1:46When a is not 1
- 2:32Your turn
- 3:10Three traps
- 3:44Recap
Lesson transcript
The narration of the video, word for word, under its chapter headings.
0:00The vertex is the whole game
Welcome to SAT Climb. On a parabola question, the vertex is the whole game, the highest or lowest point. Standard form hides it. But one move, completing the square, rewrites any quadratic into vertex form, and the vertex just falls out. No graphing calculator needed. Let's make it automatic.
0:22Why it is called completing the square
Vertex form is y equals a, times x minus h, squared, plus k. The vertex is simply h, k, read it right off. Standard form buries that. So why do we call it completing the square? Picture x squared as an actual square, and the x term as a strip you split in half and wrap around two sides. That leaves one empty corner. To fill it, you take half of b and square it. Add that corner, and the expression becomes a perfect square. That is the entire method, no matter the numbers.
1:02Worked example: x squared minus 6x plus 5
Let's do it. y equals x squared minus six x plus five. Focus on the x terms. Take half of negative six, that is negative three. Square it, you get nine. Add nine and subtract nine, so you haven't changed anything. Now x squared minus six x plus nine is a perfect square, x minus three, squared. The leftover, negative nine plus five, is negative four. So y equals x minus three, squared, minus four. The vertex is three, negative four. And notice the sign flips: x minus three means x equals positive three.
1:46When a is not 1
What if a is not one? y equals two x squared plus eight x plus three. First, factor the two out of just the x terms. Two times, x squared plus four x, plus three. Now complete the square inside. Half of four is two, squared is four, so inside we get x plus two, squared, minus four. Here is the trap. That minus four is inside the parentheses, so it gets multiplied by the two. Two times negative four is negative eight. Add the plus three, and that is negative five. Final answer, two times x plus two, squared, minus five. Vertex, negative two, negative five.
2:32Your turn
Your turn. Write y equals x squared plus four x plus one in vertex form. Pause if you need to. Half of four is two, squared is four. Add and subtract four. x squared plus four x plus four is x plus two, squared. One minus four is negative three. So y equals x plus two, squared, minus three. The vertex is negative two, negative three. Plot it, and the parabola bottoms out right there.
3:10Three traps
Three traps that cost points. One, adding a square but forgetting to subtract it. You must keep the equation balanced, add and subtract the same number. Two, when a is not one, forgetting to multiply the pulled-out constant by a. That minus four became negative eight. Three, the sign of h. x minus three, squared means the vertex sits at positive three. x plus two, squared means negative two. Always flip the sign you see inside the parentheses.
3:44Recap
Completing the square, solved. Take half of b, square it, add and subtract, then factor the perfect square. If a is not one, pull it out first. The vertex reads straight off. Start practicing free at satclimb.com. Your SAT is closer than you think.
Read the written version: the Nonlinear functions strategy guide, then try 25 hard Nonlinear functions questions with full explanations. Both are free, no account needed.