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Quadratics (Hard)

Late AMC 10/12 through early AIME — dense computation.

Easy Medium Hard AIME / USAJMO / AMO

Quadratics: core idea

Parameter ranges.

Example problem: hint and answer

Roots of $x^2-7x+10$ are $r,s$ . Find $r^2+s^2$ .

Hint: Use factoring, Vieta's relations, or the discriminant. Answer: 29.

Practice (9 problems — type your answers)

1. Find the minimum of $x^2-12x+50$ .
2. For what $k$ does $x^2+kx+16=0$ have one real root?
3. $y=x^2-6x+c$ touches the $x$ -axis. Find $c$ .
4. If $x+1/x=5$ , find $x^2+1/x^2$ .
5. Find the larger root of $2x^2-9x-5=0$ .
6. A rectangle has perimeter $30$ and area $56$ . Find its longer side.
7. Find the sum of positive roots of $x^4-13x^2+36=0$ .
8. For how many integers $n$ does $x^2-nx+12=0$ have integer roots?
9. $y=x^2+bx+4$ passes through $(2,0)$ . Find $b$ .

Quadratics Practice Problems — Hard