Apparently, I got this question wrong on a test and I'd like to know what error I made... can you help?
Here's what I did:
2b^2+b=6
b^2+1/2b=3
b^2+1/2b+(1/2)^2=3+(1/2)^2
(b+1/2)^2=3+1/4
b+1/2=+/- <13/4>
b=-1/2+/-<13>/2
b=(-1+/-<13>)/2
<> means its in a radical and +/- means plus or minus
The error is in your third line
you take 1/2 of the middle coefficient, and then square it
(1/2)(1/2) = 1/4 --- (1/4)^2 = 1/16
3rd line:
b^2 + (1/2)b + 1/16 = 3 + 1/16
(b+1/4)^2 = 49/16
b + 1/4 = ± 7/4
b = -1/4 ± 7/4 = 6/4 or -8/4
b = -2 or 3/2
since we have rational answers, we know it would have factored to
(x+2)(2b-3)
= 2b^2 + b - 6 , the original.
Sure, I can help you identify the error you made. Let's go step-by-step through your work.
Starting from the equation 2b^2 + b = 6, you correctly rearranged it to b^2 + 1/2b = 3.
Then, to complete the square, you added (1/2)^2 = 1/4 on both sides of the equation. This step is also correct.
You correctly simplified the right-hand side to 3 + 1/4.
However, there seems to be an error in the next step. You wrote (b + 1/2)^2 = 3 + 1/4, which is incorrect.
The correct expansion of (b + 1/2)^2 is b^2 + b + 1/4, not b^2 + 1/2b + 1/4.
Therefore, the correct equation should be b^2 + b + 1/4 = 3 + 1/4.
From there, you can subtract 1/4 from both sides of the equation, giving you:
b^2 + b = 3.
Now, this is a quadratic equation. To solve it, you can either factor it or use the quadratic formula.
If you decide to factor, you would rearrange it as:
b^2 + b - 3 = 0.
However, because this equation does not factor nicely, it is better to use the quadratic formula:
b = (-b ± √(b^2 - 4ac)) / 2a.
In this case, a = 1, b = 1, and c = -3. Substituting these values into the quadratic formula, you would get:
b = (-1 ± √(1^2 - 4(1)(-3))) / (2(1)).
Calculating the discriminant, you would have:
b = (-1 ± √(1 + 12)) / 2.
Simplifying further, you would obtain:
b = (-1 ± √13) / 2.
So, the correct solution to the quadratic equation 2b^2 + b = 6 is:
b = (-1 ± √13) / 2.
As a final note, make sure to double-check your calculations, especially when dealing with algebraic manipulations and solving equations.