Examkin

Question

Difficulty: HardQuadratic Equations and Factoring

The quadratic equation 2x2+px+q=02x^2 + px + q = 0, where pp and qq are constants, has roots rr and ss. The quadratic equation x2+(p−2)x+24=0x^2 + (p - 2)x + 24 = 0 has roots r+2r + 2 and s+2s + 2. What is the value of qq?

Answer: 32

Answer

The value of qq is 32.
Applying Vieta's formulas to 2x2+px+q=02x^2 + px + q = 0 gives r+s=−p/2r + s = -p/2 and rs=q/2rs = q/2. For the second equation x2+(p−2)x+24=0x^2 + (p-2)x + 24 = 0, the sum of roots is (r+2)+(s+2)=−(p−2)(r+2) + (s+2) = -(p-2), which simplifies to (r+s)+4=2−p(r+s) + 4 = 2 - p. Substituting r+s=−p/2r+s = -p/2 yields −p/2+4=2−p-p/2 + 4 = 2 - p, solving to p=−4p = -4 and r+s=2r+s = 2. The product of roots for the second equation is (r+2)(s+2)=rs+2(r+s)+4=24(r+2)(s+2) = rs + 2(r+s) + 4 = 24. Substituting rs=q/2rs = q/2 and r+s=2r+s = 2 gives q/2+4+4=24q/2 + 4 + 4 = 24, which simplifies to q/2=16q/2 = 16 and q=32q = 32.

Step-by-Step Solution

1
Express the sum and product of roots rr and ss in terms of pp and qq using Vieta's formulas.
r+s=−p2r + s = -\frac{p}{2} and rs=q2rs = \frac{q}{2}.
For a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, the sum of the roots is −ba-\frac{b}{a} and the product is ca\frac{c}{a}.
2
Relate the sum of the shifted roots (r+2)(r + 2) and (s+2)(s + 2) to the coefficients of the second quadratic equation.
(r+2)+(s+2)=−(p−2)  ⟹  (r+s)+4=2−p(r + 2) + (s + 2) = -(p - 2) \implies (r + s) + 4 = 2 - p.
The coefficient of xx in x2+(p−2)x+24=0x^2 + (p - 2)x + 24 = 0 is (p−2)(p - 2), so the sum of its roots equals −(p−2)-(p - 2).
3
Substitute r+s=−p2r + s = -\frac{p}{2} into the sum relation to determine pp.
−p2+4=2−p  ⟹  p2=−2  ⟹  p=−4-\frac{p}{2} + 4 = 2 - p \implies \frac{p}{2} = -2 \implies p = -4.
Solving the linear equation for pp yields p=−4p = -4, which means r+s=2r + s = 2.
4
Expand the product of the shifted roots (r+2)(s+2)=24(r + 2)(s + 2) = 24 and solve for qq.
rs+2(r+s)+4=24  ⟹  q2+2(2)+4=24  ⟹  q2+8=24  ⟹  q=32rs + 2(r + s) + 4 = 24 \implies \frac{q}{2} + 2(2) + 4 = 24 \implies \frac{q}{2} + 8 = 24 \implies q = 32.
Substituting rs=q2rs = \frac{q}{2} and r+s=2r + s = 2 isolates qq, yielding q=32q = 32.

Key Concept

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

Topics

Quadratic Equations and Factoring

More questions from this topic