Here is an interesting quadratic equations question. Combines some concepts of number theory.

Question

y = x^2 + bx + 256 cuts the x axis at (h, 0) and (k, 0). If h and k are integers, what is the least value of b?

A. –32

B. –256

C. –255

D. –257

E. 0

Correct Answer : Choice D. -257

The curve cuts the x axis at (h, 0) and (k, 0). So, h and k are the roots of the quadratic equation.

For a quadratic equation of the form ax^2 + bx + c = 0, the sum of the roots = -b/a

The sum of the roots of this equation is -b/1 = -b.Higher the sum of the roots lower the value of b.

For a quadratic equation of the form ax^2 + bx + c = 0, the product of roots = c/a.

Therefore, the product of the roots of this equation = 256/1 = 256.

i.e., h*k = 256

h and k are both integers.

So, h and k are both integral factors of 256.

256 can be expressed as product of two numbers in the following ways:

1 * 256

2 * 128

4 * 64

8 * 32

16 * 16

The sum of the roots is maximum when the roots are 1 and 256 and the maximum sum is 1 + 256 = 257.

The sum of the roots is minimum when the roots are -1 and -256 and the least sum is -1 - 256 = -257.

The least value possible for b is therefore -257.

Question

y = x^2 + bx + 256 cuts the x axis at (h, 0) and (k, 0). If h and k are integers, what is the least value of b?

A. –32

B. –256

C. –255

D. –257

E. 0

Correct Answer : Choice D. -257

Explanation

The curve cuts the x axis at (h, 0) and (k, 0). So, h and k are the roots of the quadratic equation.

For a quadratic equation of the form ax^2 + bx + c = 0, the sum of the roots = -b/a

The sum of the roots of this equation is -b/1 = -b.Higher the sum of the roots lower the value of b.

For a quadratic equation of the form ax^2 + bx + c = 0, the product of roots = c/a.

Therefore, the product of the roots of this equation = 256/1 = 256.

i.e., h*k = 256

h and k are both integers.

So, h and k are both integral factors of 256.

256 can be expressed as product of two numbers in the following ways:

1 * 256

2 * 128

4 * 64

8 * 32

16 * 16

The sum of the roots is maximum when the roots are 1 and 256 and the maximum sum is 1 + 256 = 257.

The sum of the roots is minimum when the roots are -1 and -256 and the least sum is -1 - 256 = -257.

The least value possible for b is therefore -257.

Labels: GMAT Number Properties, GMAT Number Theory, GMAT Problem Solving, GMAT Quadratic Equations