Completing the square is a useful method that helps us find the peak point, or vertex, of a parabola. A quadratic equation is usually written like this:
( y = ax^2 + bx + c ).
To make it easier to find the vertex, we can change it into a different form:
( y = a(x - h)^2 + k ),
where ((h, k)) is the vertex point.
Here are the steps to complete the square:
Isolate the (x) terms: Start with the equation ( y = ax^2 + bx + c ).
Factor out (a) (if (a) is not 1): This changes the equation to ( y = a(x^2 + \frac{b}{a}x) + c ).
Complete the square: Inside the parentheses, add and subtract (\left(\frac{b}{2a}\right)^2).
Rewrite the equation in vertex form.
Example: Let's look at the equation ( y = 2x^2 + 8x + 5 ):
First, we can factor out the 2: ( y = 2(x^2 + 4x) + 5 ).
Next, we complete the square:
( y = 2(x^2 + 4x + 4 - 4) + 5 )
This simplifies to
( y = 2((x + 2)^2 - 4) + 5 ).
Now, we simplify it to find the vertex:
( y = 2(x + 2)^2 - 3 ).
In this case, the vertex is the point ((-2, -3)).
Completing the square helps us find the vertex and makes it much easier to graph parabolas!
Completing the square is a useful method that helps us find the peak point, or vertex, of a parabola. A quadratic equation is usually written like this:
( y = ax^2 + bx + c ).
To make it easier to find the vertex, we can change it into a different form:
( y = a(x - h)^2 + k ),
where ((h, k)) is the vertex point.
Here are the steps to complete the square:
Isolate the (x) terms: Start with the equation ( y = ax^2 + bx + c ).
Factor out (a) (if (a) is not 1): This changes the equation to ( y = a(x^2 + \frac{b}{a}x) + c ).
Complete the square: Inside the parentheses, add and subtract (\left(\frac{b}{2a}\right)^2).
Rewrite the equation in vertex form.
Example: Let's look at the equation ( y = 2x^2 + 8x + 5 ):
First, we can factor out the 2: ( y = 2(x^2 + 4x) + 5 ).
Next, we complete the square:
( y = 2(x^2 + 4x + 4 - 4) + 5 )
This simplifies to
( y = 2((x + 2)^2 - 4) + 5 ).
Now, we simplify it to find the vertex:
( y = 2(x + 2)^2 - 3 ).
In this case, the vertex is the point ((-2, -3)).
Completing the square helps us find the vertex and makes it much easier to graph parabolas!