Completing the Square: From Standard Form to Vertex Form

0:00Standard form is a x squared plus b x plus c. Vertex form, a times x minus h squared plus k, shows the vertex, the turning point of the parabola, at h, k.

0:15As algebra tiles, x squared plus 6 x plus 5 is one x squared tile, six x tiles and five unit tiles.

0:25Halve the x tiles: three go on the right and three go below.

0:30The missing corner is 3 by 3, so 9 unit tiles complete the square.

0:36The rule: halve the coefficient of x, the number in front of x, and square it. Half of 6 is 3, and 3 squared is 9.

0:46Adding 9 changes the value, so we also subtract 9, using nine negative tiles.

0:53A positive and a negative tile make zero, so five pairs cancel, leaving negative 4.

1:00The tiles show x plus 3 squared minus 4. Since x plus 3 is x minus negative 3, the vertex is negative 3, negative 4.

1:11Now a is 2: for 2 x squared minus 12 x plus 7, factor 2 out of the x terms only.

1:20Half of negative 6 is negative 3, and its square is 9, so x squared minus 6 x is x minus 3 squared minus 9.

1:31Expanding, the 2 multiplies both parts: minus 9 becomes minus 18. Forgetting the 2 is a common mistake.

1:41Minus 18 plus 7 is minus 11, giving 2 times x minus 3 squared minus 11, with vertex 3, negative 11.

1:52The same steps solve equations: x squared plus 6 x plus 5 equals 0 becomes x plus 3 squared minus 4 equals 0, so x plus 3 squared is 4.

2:06Then x plus 3 is 2 or negative 2, so x is negative 1 or negative 5.

2:14Recap: to complete the square, factor out a, halve the coefficient of x, square it, then add and subtract it.