9.1. General Form of a Conic http://www.ck12.org
Before you start manipulating the general form of a conic equation you should be able to recognize whether it is a
circle, ellipse, parabola or hyperbola. In standard form, the two coefficients to examine areAandC.
- Forcircles,the coefficients ofx^2 andy^2 are the same sign and the same value:A=C
- Forellipses,the coefficients ofx^2 andy^2 are the same sign and different values:A,C> 0 ,A 6 =C
- Forhyperbolas,the coefficients ofx^2 andy^2 are opposite signs:C< 0 <AorA< 0 <C
- Forparabolas,either the coefficient ofx^2 ory^2 must be zero:A=0 orC= 0
Each specific type of conic has its own graphing form, but in all cases the technique of completing the square is
essential. The examples review completing the square and recognizing conics.
Example A
Complete the square in the expressionx^2 + 6 x. Demonstrate graphically what completing the square represents.
Solution: Algebraically, completing the square just requires you to divide the coefficient ofxby 2 and square the
result. In this case(^62 )^2 = 32 =9. Since you cannot add nine to an expression without changing its value, you must
simultaneously add nine and subtract nine so the net change will be zero.
x^2 + 6 x+ 9 − 9
Now you can factor by recognizing a perfect square.
(x+ 3 )^2 − 9
Graphically the original expressionx^2 + 6 xcan be represented by the area of a rectangle with sidesxand(x+ 6 ).
The term “complete the square” has visual meaning as well algebraic meaning. The rectangle can be rearranged to
be more square-like so that instead of small rectangle of area 6xat the bottom, there is a rectangle of area 3xon two