If b, c, and d are constants and x^2 + bx + c = (x + d)^2

If b, c, and d are constants and x^2 + bx + c = (x + d)^2 for all values of x, what is the value of c?

\(x^2 + bx + c = (x + d)^2\) --> \(x^2+bx+c=x^2+2dx+d^2\) --> \(bx+c=2dx+d^2\).

Now, as above expression is true "for ALL values of \(x\)" then it must hold true for \(x=0\) too --> \(c=d^2\).

Next, substitute \(c=d^2\) --> \(bx+d^2=2dx+d^2\) --> \(bx=2dx\) --> again it must be true for \(x=1\) too --> \(b=2d\).

So we have: \(c=d^2\) and \(b=2d\). Question: \(c=?\)

(1) \(d=3\) --> as \(c=d^2\), then \(c=9\). Sufficient
(2) \(b=6\) --> as \(b=2d\) then \(d=3\) --> as \(c=d^2\), then \(c=9\). Sufficient.

Answer: D.
_________________

This question is based on special algebraic equations, particularly on (x + y)^2 = x^2 + 2xy + y^2.

So (x + d)^2 = x^2 + 2xd + d^2 = x^2 + bx + c

Equating the co-efficients of x, 2d = b
and equating the constant terms, d^2 = c

If I have d = 3, I get b = 6 and c = 9. (Statement I alone is sufficient.)
If I have b = 6, I get d = 3 and then c = 9 (Statement II alone is sufficient.)

Ha, I had this question yesterday in a practice set. I chose A but when reviewed all the answers, understood why it was D. I usually only remember the answer if I reviewed the question. I hope to get a similar one in the real exam now that I know how to solve it!

Hi,

This question should be put under the DS section.

Also the question should read as:

If b, c, and d are constants and x2 + bx + c = (x + d)^2 for all values of x, what is the value of c?

(1) d = 3
(2) b = 6

The answer will be D.

As the equation is true for all the values of X.
Hence corresponding constants attached to the powers of X should be equal
x2 + bx + c = x2 + 2dx + d^2
so b=2d and c=d^2
That means if we know the value of b or d, we can find out c

Hence answer should be D.

If b, c, and d are constants and x^2 + bx + c = (x + d)^2 for all values of x, what is the value of c?
(1) d = 3
(2) b = 6

since expanding (x + d)^2 = x^2 + 2xd + D^2

hence we have
bx + c = 2xd + D^2

we need to know value of b and value of d to get the correct answer.

Can you please explain if it is other wise.

Given \(x^2 + bx + c = (x+d)^2\)
Expanding on both sides, we get: \(x^2 + 2dx + d^2 = x^2 + bx + c.\) We can cancel the x^2 on both sides and that leaves us with: \(2dx + d^2 = bx + c\)

And this is valid for all values of x, we are given. Let's just substitute two values of x:

\(x = 1\)

Then \(2d + d^2 = b + c\)

\(x = 2\)

\(4d + d^2 = 2b + c\)
Now taking these two equations together:
\(2d + d^2 = b + c\\
4d + d^2 = 2b + c\)

Multiplying the first equation by 2 and solving, we get:

\(4d + 2d^2 = 2b + 2c\\
4d + d^2 = 2b + c\)

So we get: \(d^2 = c\)

Thus, from our two statements, we know that statement one is sufficient by itself. Now look at statement 2. And look at the first equation we had: \(2d + d^2 = b + c.\) But then \(c = d^2\), which means that \(2d = b\). So, if b is given, you can find d and hence c. Thus statement 2 is also sufficient.

Thus the trick here is to read the question carefully. If it's valid for all values of x, it's valid for any two values of x. You can arbitrarily pick x. In fact, picking x = 0 might be even better and help you solve the problem much faster, in hindsight.

What if I solve it the way described below? will it be wrong?

Given x^2 + bx + c = (x + d)^2

When these two equations will equate to zero

- From (x+d)^2=0 we can determine that x will only have one unique value i.e. x=-d

- if x will have only one unique value, then from x^2 + bx + c = 0 we can say b^2 - 4ac = 0, where a=1

which means b^2 = 4c
-> c= (b^2)/4 ....(a)

also when b^2 - 4ac = 0 then
x= -b/2a (using x = (-b ±√(b^2 - 4ac))/2a )
where x=d
hence
d=-b/2
b=-2d ...(b)

using (b) on (a)

c=d ...(c)


Now,
evaluating statement (1), value of d is sufficient to determine value of c using (c)
evaluating statement (2), value of b is sufficient to determine value of c using (a)

Hence the answer is D

Is this approach correct?

The discriminant of \(x^2+bx+c\) should be zero -> \(b^2-4c=0\)
Equating the roots of expression \(-b/2=-d\)
Hence if either b or d is known c can be calculated.

If b, c, and d are constants and x^2 + bx + c = (x + d)^2 for all values of x, what is the value of c?

(1) d = 3
(2) b = 6

---------------------------------------------------------------------------------------------------------------------------------------------------------

x^2 + bx + c = (x + d)^2 ...................... (1)

using equation 1 , we can say : x^2 + bx + c can be converted into whole square only when b^2 - 4ac = 0 OR -b/2a = -d

Statement 1 :

d = 3 ; also we know a=1

-b/2a = -d
-b/2(1) = -(3)
b=6

b^2-4ac=0
(6)^2 - 4(1)(c) = 0
c=9

sufficient.

Statement 2

b=6

b^2-4ac=0
(6)^2 - 4(1)(c) = 0
c=9

sufficient.

Answer D


Experts pls comment.

Responding to a pm:


Equate the co-efficients as shown in my post above:
https://gmatclub.com/forum/if-b-c-and-d ... ml#p802314

Depending on what is asked and what is given, you can answer the question.

thanks mam

Hi brunel, in statement 2, why is b=2d is not 2x3=6 but is d^2 > 9 ? Am I missing something here? Thanks

No worry now brunel, I work it out. b=2d > 6=2d > d=3. Thanks

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________