Question

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

Answer 1

`(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.

Answer 2

`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`

Answer 3

`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`