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AMC 10 & 12

Quadratics & Parabolas

The quadratic formula, the discriminant, and vertex form — three views of the same parabola that each answer a different question fast.

A quadratic ax2+bx+c=0ax^2+bx+c=0 (with a≠0a \neq 0) always has exactly two roots, counted with multiplicity and allowing complex values. The quadratic formula gives them directly:

x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}

The discriminant

The expression under the square root, Δ=b2−4ac\Delta = b^2 - 4ac, tells you the nature of the roots before you compute them:

  • Δ>0\Delta > 0: two distinct real roots
  • Δ=0\Delta = 0: one repeated real root (the parabola is tangent to the x-axis)
  • Δ<0\Delta < 0: two complex conjugate roots (the parabola never touches the x-axis)

This is enormously useful on problems that ask “for what values of kk does this equation have exactly one solution” — you never need to solve the equation at all, just set Δ=0\Delta=0and solve for kk.

Vertex form

Completing the square turns y=ax2+bx+cy=ax^2+bx+c into vertex form, which reads off the parabola's turning point directly:

y=a(x−h)2+k,h=−b2a,k=c−b24ay = a(x-h)^2 + k, \qquad h = -\dfrac{b}{2a},\quad k = c - \dfrac{b^2}{4a}

The vertex is (h,k)(h, k), and x=hx=h is the axis of symmetry. If a>0a>0 the vertex is a minimum; if a<0a<0 it's a maximum — which makes vertex form the fastest route to any “maximize/minimize this quadratic expression” problem.

Worked example

Problem: Find all values of kk for which x2+kx+(k+3)=0x^2 + kx + (k+3) = 0has exactly one real solution.

Solution: “Exactly one real solution” means the discriminant is zero:

Δ=k2−4(k+3)=0  ⟹  k2−4k−12=0\Delta = k^2 - 4(k+3) = 0 \;\Longrightarrow\; k^2 - 4k - 12 = 0

This factors as (k−6)(k+2)=0(k-6)(k+2)=0, so k=6k = 6 or k=−2k = -2. Notice this second equation in kk is itself a quadratic — parameter problems often just push the discriminant one level up.

Practice problems

1.Without solving, determine the number of real roots of 2x2−3x+5=02x^2 - 3x + 5 = 0.

2.Find the vertex of y=2x2−8x+3y = 2x^2 - 8x + 3.

3.For what value(s) of mm does x2−6x+m=0x^2 - 6x + m = 0 have two distinct real roots?

4.What is the minimum value of f(x)=3x2+12x+7f(x) = 3x^2 + 12x + 7?

Absolute ValuePolynomial Theory