If the quadratic equation x^2 - kx + 36 = 0 has two distinct positive integer roots, r_1 a… — GMAT

If the quadratic equation $x^2 - kx + 36 = 0$ has two distinct positive integer roots, $r_1$ and $r_2$ , and $r_1 + 2$ and $r_2 + 2$ are the roots of the quadratic equation $x^2 - 19x + m = 0$ , what is the value of $m$ ?

Answer: 70

Answer

The value of

m

is 70.

By applying Vieta's formulas to both quadratic equations, we find that the sum of the original roots is

r_1 + r_2 = 15

and their product is

r_1 r_2 = 36

. Factoring

x^2 - 15x + 36 = (x-3)(x-12) = 0

confirms the original roots are

3

and

12

. The new roots are

3+2=5

and

12+2=14

. Their product

m = 5 \times 14 = 70

Step-by-Step Solution

Apply Vieta's relations to the first quadratic equation

r_1 + r_2 = k

and

r_1 \cdot r_2 = 36

For any quadratic equation

x^2 + bx + c = 0

, the sum of roots is

-b

and the product of roots is

c

Apply Vieta's relations to the second quadratic equation

(r_1 + 2) + (r_2 + 2) = 19 \implies r_1 + r_2 = 15

The coefficient of

x

x^2 - 19x + m = 0

dictates that the sum of its roots equals

19

Determine the roots

r_1

and

r_2

and check integer constraints

r_1 = 3

and

r_2 = 12

The integer factors of

36

that sum to

15

are

3

and

12

, satisfying all conditions.

Compute the constant term

m

for the transformed equation

m = (3 + 2)(12 + 2) = 5 \times 14 = 70

The constant term

m

equals the product of the transformed roots

(r_1 + 2)

and

(r_2 + 2)

Key Concept

Vieta's Formulas and Quadratic Root Transformations

Estimated Time:2m 0s

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