Take Note Key Concept: Completing the Square

You can form a perfect square trinomial from x 2 + b x x squared , plus b x  by adding ( b 2 ) 2 . open , b over 2 , close squared . .

x 2 + b x + ( b 2 ) 2 = ( x + b 2 ) 2 x squared , plus b x plus . open , b over 2 , close squared . equals . open . x plus , b over 2 . close squared

Problem 4 Completing the Square

What value completes the square for x 2 − 10 x ? bold italic x squared , minus 10 bold italic x question mark  Justify your answer.

x 2 − 10 x Identify  b ; b = − 10 ( b 2 ) 2 = ( − 10 2 ) 2 = ( − 5 ) 2 = 25 Find ( b 2 ) 2 . x 2 − 10 x + 25 Add the value of  ( b 2 ) 2 to complete the square . x 2 − 10 x + 25 = ( x − 5 ) 2 Rewrite as the square of a binomial . table with 4 rows and 2 columns , row1 column 1 , x squared , minus 10 x , column 2 cap identify . b semicolon b equals negative 10 , row2 column 1 , open , b over 2 , close squared . equals . open , negative 10 over 2 , close squared . equals . open , negative 5 , close squared . equals 25 , column 2 cap find . open , b over 2 , close squared . . , row3 column 1 , x squared , minus 10 x plus 25 , column 2 cap addthevalueof . open , b over 2 , close squared . tocompletethesquare . . , row4 column 1 , x squared , minus 10 x plus 25 equals . open , x minus 5 , close squared , column 2 cap rewriteasthesquareofabinomial . . , end table

✓ Got It?

4. 1. What value completes the square for x 2 + 6 x ? x squared , plus 6 x question mark
   2. Reasoning Is it possible for more than one value to complete the square for an expression? Explain.

Take Note Key Concept: Solving an Equation by Completing the Square

1. Rewrite the equation in the form x 2 + b x = c . x squared , plus b x equals c .  To do this, get all terms with the variable on one side of the equation and the constant on the other side. Divide all the terms of the equation by the coefficient of x 2 x squared  if it is not 1.
2. Complete the square by adding ( b 2 ) 2 open , b over 2 , close squared  to each side of the equation.
3. Factor the trinomial.
4. Find square roots.
5. Solve for x.