QUADRATICS

Overview

A quadratic equation is any equation you can write as $ax^2 + bx + c = 0$ with $a \neq 0$. Quadratics appear constantly on the AMC10 — sometimes as the main event, sometimes hiding inside a word problem or a system of equations. The key insight is that you rarely need to find the roots explicitly; most AMC questions ask for sums, products, or expressions in terms of the roots, which Vieta's formulas deliver in a single step.

Quadratics also connect to parabolas ($y = ax^2 + bx + c$), where completing the square reveals the vertex and minimum/maximum value instantly. Knowing when to factor, when to use the quadratic formula, and when to reach for Vieta's or the vertex form is the art of quadratic fluency.

The discriminant $\Delta = b^2 - 4ac$ controls the number of real roots: positive means two distinct real roots, zero means one repeated root, negative means no real roots. AMC problems often hinge on setting $\Delta$ equal to zero or asking how many values of a parameter give real solutions.

Key facts

• Standard form: $ax^2 + bx + c = 0$, $a \neq 0$.
• Quadratic formula: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ — use when factoring isn't obvious.
• Vieta's sum of roots: $r_1 + r_2 = -\dfrac{b}{a}$ — reaches the answer without finding individual roots.
• Vieta's product of roots: $r_1 r_2 = \dfrac{c}{a}$ — same idea for products.
• Discriminant: $\Delta = b^2 - 4ac$. $\Delta > 0$ → two real roots; $\Delta = 0$ → one repeated real root; $\Delta < 0$ → no real roots.
• Vertex form: $y = a(x - h)^2 + k$, vertex at $(h, k)$. Minimum value $k$ when $a > 0$; maximum when $a < 0$.
• Completing the square: $ax^2 + bx + c = a\!\left(x + \dfrac{b}{2a}\right)^{\!2} + c - \dfrac{b^2}{4a}$.
• Identity toolkit: $r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2$ — express symmetric functions of roots using Vieta values alone.

Worked example 1

The roots of $x^2 - 8x + 7 = 0$ are $r$ and $s$. Find $r^2 + s^2$.

We want $r^2 + s^2$, but we don't need to find $r$ and $s$ individually. By Vieta's formulas:

$r + s = \frac{-(-8)}{1} = 8, \qquad rs = \frac{7}{1} = 7.$

Now use the identity $r^2 + s^2 = (r+s)^2 - 2rs$:

$r^2 + s^2 = 8^2 - 2(7) = 64 - 14 = 50.$

(Quick check: the roots are $1$ and $7$, so $r^2 + s^2 = 1 + 49 = 50$. ✓)

Worked example 2

Find all values of $m$ such that $x^2 - (2m-1)x + (m^2 - m - 2) = 0$ has a repeated real root. What is the sum of all such values of $m$?

A repeated root occurs when the discriminant equals zero:

$\Delta = (2m-1)^2 - 4(m^2 - m - 2) = 0.$

Expand:

$4m^2 - 4m + 1 - 4m^2 + 4m + 8 = 0 \implies 9 = 0.$

That's a contradiction — there is no value of $m$ giving a repeated root for this particular equation. This illustrates that setting $\Delta = 0$ can produce zero, one, or two solutions depending on how the coefficients depend on the parameter.

For a version that does have solutions, consider $x^2 - (m+3)x + (3m + 4) = 0$. Setting $\Delta = 0$:

$(m+3)^2 - 4(3m+4) = 0 \implies m^2 - 6m - 7 = 0 \implies (m-7)(m+1) = 0.$

So $m = 7$ or $m = -1$, and their sum is $\boxed{6}$.

Common traps

• Forgetting the sign in Vieta's sum. The sum of roots is $-b/a$, not $+b/a$. If $x^2 + 5x + 6 = 0$, the sum is $-5$, not $5$.
• Confusing "equal roots" with "integer roots." Equal roots means $\Delta = 0$. Integer roots means $\Delta$ is a perfect square and the formula gives whole numbers — a stricter condition.
• Dropping the leading coefficient. Vieta gives $r_1 + r_2 = -b/a$ and $r_1 r_2 = c/a$. When $a \neq 1$, students often forget to divide by $a$.
• Vertex vs. root. The vertex $x = -b/(2a)$ is not a root (unless the vertex touches the $x$-axis). Don't mix up the vertex formula with the quadratic formula.
• Assuming two solutions always exist. Check $\Delta$ before claiming roots. A problem may intend the "no real solution" case as a trap answer.