How do you solve #49b^2 + 84b + 32 = 0 # by completing the square?

3 Answers
Jan 24, 2017

Answer:

#(7b + 6)^2 + 32 - 36 = 0#
#(7b + 6)^2 = 4#
#7b + 6 = +-2#
#7b = -4# or #7b = -8#
#b = -4/7, -8/7#

Explanation:

The first step is to find the number that squares to give 49, which is 7. Then, you need to find a number such that #7 * x = 84/2#, so that when you expand the bracket, you get two terms of #42b# which added together gives #84b#. #7 * 6 = 42#, so 6 is the number we want. Now, putting that into the bracket works fine, but you have to account for the fact that when expanded it will give an extra #6 * 6 = 36#, so subtract 36 outside the bracket.

After that, it's a case of square rooting both sides, then rearranging to get the answer. Since both #2^2# and #(-2)^2# give 4, you arrive at 2 different answers, both of which are valid.

Jan 24, 2017

Answer:

#b = -4/7" "# or #" "b = -8/7#

Explanation:

The difference of squares identity can be written:

#A^2-B^2=(A-B)(A+B)#

We will use this with #A=7b# and #B=6#...

Note that #49 = 7^2#, #84/(2*7) = 6# and #6^2 = 36#

So we find:

#0 = 49b^2+84b+32#

#color(white)(0) = (7b)^2+2(7b)(6)+(6)^2-4#

#color(white)(0) = (7b)^2+2(7b)(6)+(6)^2-4#

#color(white)(0) = (7b+6)^2-2^2#

#color(white)(0) = ((7b+6)-2)((7b+6)+2)#

#color(white)(0) = (7b+4)(7b+8)#

Hence:

#b = -4/7" "# or #" "b = -8/7#

Jan 25, 2017

Answer:

# b = -8/7, -4/7#

Explanation:

This is the method I was taught to complete the square on a general quadratic:

# x^2+bx+c #

  • Step 1: Factor out (or divide) the coefficient of #x^2# so that that coefficient is #1#.
  • Step 2:Use the knowledge of a perfect square #(x+alpha)^2=x^2+2alphax+alpha^2#, Here we have #2alpha=b=>alpha=1/2b# and subtract #alpha^2=(1/2b)^2#
  • Step 3: Solve the equation

So for this particular problem we have

Step 1: Divide by #a=49#

# 49b^2 + 84b + 32 = 0 #
# :. b^2 + 84/49b + 32/49 = 0 #
# :. b^2 + 12/7b + 32/49 = 0 #

Step 2: Form a perfect square using #(x+b/2)^2#

# :. (b + 1/2*12/7)^2 - (1/2*12/7)^2 + 32/49=0 #
# :. (b + 6/7)^2 - (6/7)^2 + 32/49=0 #
# :. (b + 6/7)^2 - 36/49 + 32/49=0 #
# :. (b + 6/7)^2 - 4/49=0 #

Step 3:: If we are solving an equation then solve it

# :. (b + 6/7)^2 = 4/49 #
# :. b + 6/7 = +-sqrt(4/49) #
# :. b + 6/7 = +-2/7 #
# :. b = - 6/7 +-2/7 #

Leading to the two solutions:

# b = - 6/7 -2/7 = -8/7#

# b = - 6/7 +2/7 = -4/7#