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Quadratics (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

Easy Medium Hard AIME / USAJMO / AMO

Quadratics: core idea

Discriminant.

Example problem: hint and answer

Solve $x^2-7x+12=0$ for the larger root.

Hint: Complete the square or apply the quadratic formula when needed. Answer: 4.

Practice (9 problems — type your answers)

1. Find the discriminant of $x^2+4x-5=0$ .
2. Find the minimum value of $x^2-8x+3$ .
3. Solve $2x^2-8x=0$ for the nonzero root.
4. If $x^2-5x+6=0$ , find $x^2$ for the larger root.
5. Find the positive root of $x^2-2x-8=0$ .
6. For $y=-x^2+6x$ , find the maximum $y$ -value.
7. Solve $x^2=6x-5$ for the smaller root.
8. Find $k$ if $x^2+kx+9$ has double root $x=-3$ .
9. Find the $y$ -intercept of $y=(x-2)(x+5)$ .

Quadratics Practice Problems — Medium