Circles

Circle Complete the Square

Convert a circle from expanded form to standard form by completing the square in x and y.

circle-complete-the-square Complete the square for each variable, then read the center and radius from standard form.

Example

Complete the square to find the circle center and radius.

highlighted = computed this step

Step 1 — Group terms

Group the x and y terms.

x2−4x+y2+6y=12x^{2} -4 x + y^{2} + 6 y = 12

Step 2 — Complete the x-square

Complete the square for x.

x2−4x+4=(x−2)2x^{2} -4 x + \hl{4} = (x- \hl{2} )^{2}

Step 3 — Complete the y-square

Complete the square for y.

y2+6y+9=(y+3)2y^{2} + 6 y + \hl{9} = (y+ \hl{3} )^{2}

Step 4 — Add to both sides

Add the completed-square constants.

12+4+9=2512 + 4 + 9 = \hl{25}

Step 5 — Standard form

Write the standard form.

(x−2)2+(y+3)2=25(x- \hl{2} )^{2} + (y+ \hl{3} )^{2} = \hl{25}

Step 6 — Plot the circle

Plot the completed circle.

Coordinate plot\text{Coordinate plot}
Completed-square circle plotCircle centered at C(2, -3) with radius 5.C

Step 7 — Center and radius

State the center and radius.

C=(2,-3)r=5C=( \hl{2} , \hl{-3} ) \quad r= \hl{5}