Multiple choice

I: If $x+2$ is factor of $x^{2}+2ax+b$ and $x^{2}+2cx+d$, then $\dfrac{b-d}{a-c}=4$ II: The equation whose roots are the squares of the roots of $x^{2}+4x+5=0$ is $x^{2}-6x+25=0$ Which of the above statement(s) is/are true?

  1. Only I

  2. Only II

  3. Both I and II

  4. Neither I nor II
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For I: If x+2 is a factor, then f(-2)=0. So 4-4a+b=0 and 4-4c+d=0. Thus b=4a-4 and d=4c-4. (b-d)/(a-c) = (4a-4c)/(a-c) = 4. For II: If roots of x^2+4x+5=0 are alpha, beta, then alpha+beta=-4 and alpha*beta=5. The new roots are alpha^2, beta^2. Sum = (alpha+beta)^2 - 2alpha*beta = 16-10=6. Product = (alpha*beta)^2 = 25. Equation is x^2-6x+25=0. Both are true.

AI explanation

For statement I, if x + 2 is a factor of both polynomials, then x = -2 makes both expressions zero, giving 4 - 4a + b = 0 and 4 - 4c + d = 0. This means b = 4a - 4 and d = 4c - 4, so (b - d) divided by (a - c) equals (4a - 4c) divided by (a - c), which is 4. For statement II, if the original roots are alpha and beta, the new roots are their squares, making the sum alpha squared + beta squared, which equals (alpha + beta) squared - 2(alpha)(beta), or 16 - 10 = 6; the product is (alpha(beta)) squared, or 25, confirming the new equation is x squared - 6x + 25 = 0.

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