If -6 and 3 are the solutions of the...

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BTGmoderatorLU: Moderator; Posts: 2544; Joined: Sun Oct 15, 2017 1:50 pmFollowed by: 6 members

by BTGmoderatorLU » Tue Dec 05, 2017 11:30 am

If -6 and 3 are the solutions to the equation x^2+bx+c=0, where b and c are constants, what is the value of b+c?

A. -18
B. -15
C. -3
D. 3
E. 8

The OA is B.

I'm confused with this PS question. Please, can any expert assist me with it? Thanks in advanced.

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Source: Beat The GMAT — Problem Solving | Tue Dec 05, 2017 11:30 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Tue Dec 05, 2017 12:31 pm

Given xÂ² + bx + c = 0:
-b = the SUM of the two solutions.
c = the PRODUCT of the two solutions.

LUANDATO wrote:If -6 and 3 are the solutions to the equation x^2+bx+c=0, where b and c are constants, what is the value of b+c?

A. -18
B. -15
C. -3
D. 3
E. 8

-b = the SUM of the two solutions:
-b = -6 + 3
-b = -3
b = 3.

c = the PRODUCT of the two solutions:
c = -6*3
c = -18.

Thus:
b+c = 3 + (-18) = -15.

The correct answer is B.

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Tue Dec 05, 2017 5:43 pm

Another way:

Remember that the roots of (x + r) * (x + s) = 0 are x = -r and x = -s.

We're told that -r = -6 and -s = 3, so r = 6 and s = -3. Plugging those in, we have

(x + 6) * (x + -3) = 0

and foiling

x*x + 6x - 3x - 18 = 0

or

x*x + 3x - 18 = 0

so b = 3, c = -18, and b + c = -15.

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Tue Dec 05, 2017 6:02 pm

Yet another way:

If our solutions are -6 and 3, then we can say

(-6)Â² -6 * b + c = 0

and

3Â² + 3b + c = 0

so:

36 - 6b + c = 0
and
9 + 3b + c = 0

Now just solve the two equations! Let's multiply the bottom one by 2:

18 + 6b + 2c = 0

then add it to the top one:

54 + 3c = 0
54 = -3c
-18 = c

Now plug c = -18 into 9 + 3b + c = 0 to solve for b:

9 + 3b - 18 = 0
3b = 9
b = 3

and we're done!

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8138; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Sat Oct 12, 2019 3:52 pm

BTGmoderatorLU wrote:If -6 and 3 are the solutions to the equation x^2+bx+c=0, where b and c are constants, what is the value of b+c?

A. -18
B. -15
C. -3
D. 3
E. 8

The OA is B.

I'm confused with this PS question. Please, can any expert assist me with it? Thanks in advanced.

Since the zeroes are x = -6 or x = 3, we know we can factor the expression x^2 + bx + c as (x + 6)(x - 3), and we can create the equation:

(x + 6)(x - 3) = x^2 + bx + c

x^2 + 3x - 18 = x^2 + bx + c

Thus, we see that b = 3 and c = -18, and their sum is -15.

Answer: B

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