For integers $b$ and $c$, what is the sum of all unique

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 amFollowed by: 3 members

by VJesus12 » Sat Nov 30, 2019 5:32 am

For integers $b$ and $c$, what is the sum of all unique solutions to the equation $x^2âˆ’bx+c=10x^2âˆ’bx+c=10?$

(1) $c = 59$
(2) $b = 14$

[spoiler]OA=C[/spoiler]

Source: Veritas Prep

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Source: Beat The GMAT — Data Sufficiency | Sat Nov 30, 2019 5:32 am

deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pmFollowed by: 5 members

by deloitte247 » Sun Dec 08, 2019 8:36 am

$$x^2-bx+c=10$$
$$x^2-bx+\left(c-10\right)=0$$
Sum of the roots of a quadractic equation = -b/a
Where b=b, and a=1
Sum = -b/1
Question=> What is the sum of all unique solutions?
Sum of roots = -b/a
Product of roots = c/a
We are looking for the sum of roots that provides a solution to the quadratic equation and answers -b/a.
Statement 1: c = 59
Remember that the solution of a quadratic equation is called its roots.
Therefore, sum of roots = -b/a, where a=1, b=?
Since'b' is not known, statement 1 is NOT SUFFICIENT.

Statement 2: b=14
Sum or roots = -14/1 = -14
Statement 2 alone is SUFFICIENT.

Answer is option B.

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