The quadratic equation 2x^2 + px + q = 0, where p and q are constants, has roots r and s. … — GRE General Test

The quadratic equation $2x^2 + px + q = 0$ , where $p$ and $q$ are constants, has roots $r$ and $s$ . The quadratic equation $x^2 + (p - 2)x + 24 = 0$ has roots $r + 2$ and $s + 2$ . What is the value of $q$ ?

Cevap: 32

Cevap

The value of

q

is 32.

Applying Vieta's formulas to

2x^2 + px + q = 0

gives

r + s = -p/2

and

rs = q/2

. For the second equation

x^2 + (p-2)x + 24 = 0

, the sum of roots is

(r+2) + (s+2) = -(p-2)

, which simplifies to

(r+s) + 4 = 2 - p

. Substituting

r+s = -p/2

yields

-p/2 + 4 = 2 - p

, solving to

p = -4

and

r+s = 2

. The product of roots for the second equation is

(r+2)(s+2) = rs + 2(r+s) + 4 = 24

. Substituting

rs = q/2

and

r+s = 2

gives

q/2 + 4 + 4 = 24

, which simplifies to

q/2 = 16

and

q = 32

Adım Adım Çözüm

Express the sum and product of roots

r

and

s

in terms of

p

and

q

using Vieta's formulas.

r + s = -\frac{p}{2}

and

rs = \frac{q}{2}

For a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

and the product is

\frac{c}{a}

Relate the sum of the shifted roots

(r + 2)

and

(s + 2)

to the coefficients of the second quadratic equation.

(r + 2) + (s + 2) = -(p - 2) \implies (r + s) + 4 = 2 - p

The coefficient of

x

x^2 + (p - 2)x + 24 = 0

(p - 2)

, so the sum of its roots equals

-(p - 2)

Substitute

r + s = -\frac{p}{2}

into the sum relation to determine

p

-\frac{p}{2} + 4 = 2 - p \implies \frac{p}{2} = -2 \implies p = -4

Solving the linear equation for

p

yields

p = -4

, which means

r + s = 2

Expand the product of the shifted roots

(r + 2)(s + 2) = 24

and solve for

q

rs + 2(r + s) + 4 = 24 \implies \frac{q}{2} + 2(2) + 4 = 24 \implies \frac{q}{2} + 8 = 24 \implies q = 32

Substituting

rs = \frac{q}{2}

and

r + s = 2

isolates

q

, yielding

q = 32

Anahtar Kavram

Relating roots and coefficients of quadratic equations using Vieta's formulas and algebraic expansion.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru