300 tough Algebra

Anurag,

It seems your approach is very easy if we understand the concept. Can you explain it again in detail?

Anurag@Gurome wrote:

yellowho wrote:What is the sum of all possible solutions for x of the equation x ( x − k ) = k + 1?

(A) 0
(B) 1
(C) k
(D) k + 1
(E) 2k – 1

Any quadratic equations in x with two roots a and b can be expressed as, (x - a)(x - b) = 0. Which on expansion takes the form, x² - (a + b)x - ab = 0

Hence, the sum of the roots = -(coefficient of x in the expanded form)

Now, x(x − k) = k + 1
=> x² - kx - (k + 1) = 0

Hence, sum of the roots = -(-k) = k

The correct answer is C.

Sum of the roots of the equation ax^2 + bx + c = 0
is -b/a

Product of the roots = c/a

roots of such an equation are

(-b + rootof(b^2 - 4ac)) / 2a

and

(-b - rootof(b^2 - 4ac)) / 2a

HTH

karthikpandian19 wrote:Anurag,

It seems your approach is very easy if we understand the concept. Can you explain it again in detail?

yellowho wrote:What is the sum of all possible solutions for x of the equation x ( x − k ) = k + 1?

(A) 0
(B) 1
(C) k
(D) k + 1
(E) 2k – 1

For any quadratic in the form ax² + bc + c = 0:
The sum of the roots = -b/a.
The product of the roots = c/a.

When x(x−k) = k + 1 is rephrased as x² - kx - (k+1) = 0:
a = the coefficient of x² = 1.
b = the coefficient of x = -k.
c = -(k+1).
Thus, the sum of the roots = -b/a = -(-k)/1 = k.

The correct answer is C.

(C) k
x= (k+(k+2))/2,(k-(k+2))/2

x= k+1,-1

hence the sum of all possible solutions of x is,
S= k+1 +(-1)
S= k

I guess its a sitter.Degree of equation is 2, so simply find out sum of roots.
Thanks

One note of caution: any question easily solved with Vieta's formulas for the sum and product of the roots of a polynomial is not typical of the GMAT. (The GMAT confines itself to Algebra 1, and Vieta is Algebra 2, at the earliest.) This question feels more like one you'd see on the Indian CAT or around the beginning of the AMC 10.

On top of that, this question takes ten seconds with Vieta, violating (something I've found to be) an unwritten GMAT rule: "If a problem can be solved with supra-GMAT math (e.g. algebra 2, trig, calculus), the supra-GMAT solution somehow takes longer than some nifty GMAT-level solution." So don't lose sleep over it!