Using Completing the Square to put equations into standard form Think Back to Geometry... 2 2_2 The standard equation of a circle with center (0,0) and radius r is x + y r x2 + y2 = 4 Center (0. 0} radius =2 . .1 /* \\J The standard equation ot a circle with center (h,k) and radius r is [x-h)2 + {y-k):2 = r2 [x-3)2 + (y+2)2 = 4 Center (3, -2) radius :2 Complete the Square to find the standard form of Circles and Quadratics CIRCLES Completing the Square will allow us to lind the standard form for the equation of a circle. Suppose you are given an equation x2 + 33- 8x- 2y+15 : 0. If you have two variables then you will need to Complete the Square twice (once with each set of variables) x2 + y2- 8x- 2y+15 : o (x2 - 8x ) + (yz- 2y )2 -15 First. group "like variables" on the left and the constants on the right [x2 - 8): +15) + ( yz- 2y +1 )= -15 +16 +1 Complete the square in each Parentheses {x - =02 +{y-1)2 = 2 Simplify This equation matches the standard form of a circle. Therefore, it is a circle with radius and center (4. 1) and radius QUADRATICS (PARABOLAS) QUADRATICS (PARABOLAS) The standard vertex form for a parabola is y: a(x-h)2+ k (h, k) represents the vertex Notice the x and y are both squared in the circle equation above. If there is only 1 variable squared then there is a slightly different process for completing the square. You will also realize that it can't be a circle if only 1 variable is squared. Use Completing the Square to rewrite the quadratic equation in general form y: x2-18x - 27 y + 27: x2-18x Take constant to the other side of the equation y + 27 + = x2-18x + Prepare to Complete the Square y + 27 +81 : x2-18x+81 Complete the Square y +103: (x-9)2 Simplify y = (x-9)2-108 Move Constant back to tinalize in the general or vertex form Vertex is (9,-108) You will also realize that it can't be a circle if only 1 variable is squared. Notice how the given equation is different than the other given parabola equation. There is only 1 variable squared but 2 variables to the first degree Example: y2+6y-8x - 31= 0 y2+6y= 8x+31 Move y terms to the same side (isolate the y terms since that is the squared term) y2+6y = 8x+31 Prepare to Complete the Square y2+6y +9 = 8x+31 + 9 Complete the square (y+3)2= 8x+ 40 Simplify ( y+3) 2= 8(x+5) Equation is now in Standard form Example: x2-4x-4y + 12= 0 x2-4x= 4y-12 Isolate the x terms x2-4x = 4y-12 Prepare to Complete the Square x2-4x+4 = 4y-12+4 Complete the Square (x-2) 2 = 4y-8 Simplify (x-2)2 = 4(y-2) Factor out the 4 from the right side. Equation now in Standard Form