In the quadratic equation x^2 + px + q = 0, p and q are prime numbers. If the equation has… — SAT

In the quadratic equation $x^2 + px + q = 0$ , $p$ and $q$ are prime numbers. If the equation has two distinct integer roots, what is the value of $p + q$ ?

5Cevap
B
6
C
7
D
9

Cevap

The correct answer is

5

. By Vieta's formulas, the sum of the roots is

-p

and the product of the roots is

q

. Since the roots are integers and

q

is prime, the roots must be

\{-1, -q\}

\{1, q\}

. The case of positive roots leads to

p + q = -1

, which is impossible for prime numbers. The case of negative roots leads to

p - q = 1

. The only prime numbers that differ by

1

are

3

and

2

. Therefore,

p = 3

and

q = 2

, and their sum is

5

Adım Adım Çözüm

Apply Vieta's formulas to express the sum and product of the roots in terms of the coefficients.

For the equation

x^2 + px + q = 0

with distinct integer roots

r

and

s

, we have

r + s = -p

and

rs = q

Vieta's formulas state that for any quadratic equation

x^2 + bx + c = 0

with roots

r

and

s

r+s = -b

and

rs = c

Analyze the product equation

rs = q

using the properties of prime numbers.

Since

q

is prime, its only integer factors are

\pm 1

and

\pm q

. Thus, the integer roots

r

and

s

must be either

\{-1, -q\}

\{1, q\}

A prime number has no positive integer divisors other than 1 and itself.

Evaluate the first case where the roots are

1

and

q

r = 1

and

s = q

, then

r + s = 1 + q = -p

, which simplifies to

p + q = -1

. Since

p

and

q

are prime numbers, they must be positive (

p, q \ge 2

). Therefore,

p + q = -1

has no solution.

Prime numbers are positive integers greater than 1, so their sum cannot be negative.

Evaluate the second case where the roots are

-1

and

-q

r = -1

and

s = -q

, then

r + s = -1 - q = -p

, which simplifies to

p - q = 1

Substituting the negative roots into the sum equation yields a positive relationship between

p

and

q

Find the prime numbers

p

and

q

that satisfy

p - q = 1

The only consecutive prime numbers are

2

and

3

. Therefore,

q = 2

and

p = 3

. Both are prime numbers, and the equation

x^2 + 3x + 2 = 0

has distinct integer roots

-1

and

-2

Since all primes except 2 are odd, any two primes with a difference of 1 must include the only even prime, 2.

Calculate the sum of

p

and

q

The sum

p + q = 3 + 2 = 5

To answer the question, we add the two identified prime values.

Anahtar Kavram

Using Vieta's formulas and number theory properties of prime numbers to solve for coefficients of a quadratic equation.

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