Inspired by Dev Sharma.

Algebra Level 5

$f(x)$ is a monic quadratic polynomial such that $f(1)=7+b$ and $f(2)=16+b$ .

And $p(x)$ is another monic quadratic polynomial such that $p(1)=4+b$ and $p(2)=7+2b$ .

If $f(x)$ and $p(x)$ have a root in common then find the product of all possible values of $b$ .

Let $x$ be the answer, then submit it as $\left | x \right |$ .

This problem is original.

: Inspiration

The answer is 117.

1 solution

Rishabh Jain
Jan 22, 2016

Let f(x)= $x^2+mx+n$ and p(x)= $x^2+px+q$ . Using f(1)=7+b and f(2)=16+b, we get $\color{goldenrod}{f(x)=x^2+6x+b}$ . Similarly using p(1)=4+b and p(2)=7+2b, we get $\color{goldenrod}{p(x)=x^2+bx+3}$ Now solving p(x) and f(x) using $~Cross ~Multiplication~Method~$ and assuming common root $\alpha$ . $\dfrac{\alpha^2}{18-b^2} = \dfrac{\alpha}{b-3} = \dfrac{1}{b-6}$ $\Rightarrow \alpha= \dfrac{b-3}{b-6} = \dfrac{18-b^2}{b-3}$ $\Rightarrow (b-3)^2=(b-6)(18-b^2)$ $\Rightarrow b^3-5b^2-24b+117=0$ Product of whose roots is -117 and hence answer=|-117|= $\Large 117$ .

Nice solution

A Former Brilliant Member - 5 years, 4 months ago

Inspired by Dev Sharma.

This problem is original.

The answer is 117.

1 solution

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