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Algebra I to AP Precalculus · Module 8 of 14

Quadratic Functions & Equations

Graphing parabolas and three methods for solving quadratic equations.

In this module
  1. Graphing Quadratics & Key Features
  2. Solving Quadratics by Factoring
  3. The Quadratic Formula
  4. Completing the Square
  5. Check yourself

Graphing Quadratics & Key Features

  • The graph of y = ax² + bx + c is a parabola: it opens upward if a > 0, downward if a < 0.
  • The axis of symmetry is x = −b/(2a); the vertex lies on this line.
  • The y-intercept is always (0, c).
Key features of a parabola
xyaxis x = 2zerozerovertex (2, −1)y-int 3
y = x² − 4x + 3 = (x − 1)(x − 3): zeros at x = 1 and 3, vertex (2, −1) on the axis of symmetry x = 2, and y-intercept 3.
Worked example

Find the axis of symmetry and the vertex for y = x² − 4x + 1.

Solution. Axis: x = −(−4)/(2·1) = 2. Vertex y-value: plug in x = 2: 4 − 8 + 1 = −3. Vertex: (2, −3).

Solving Quadratics by Factoring

  • Set the equation equal to 0, factor the quadratic expression, then apply the zero product property: if AB = 0, then A = 0 or B = 0.
  • Solve each resulting linear equation to get the two roots (x-intercepts).
Worked example

Solve x² − 5x + 6 = 0.

Solution. Factor: (x − 2)(x − 3) = 0. Zero product: x − 2 = 0 or x − 3 = 0. Solutions: x = 2 or x = 3.

The Quadratic Formula

  • For ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / (2a) — works for any quadratic, even ones that don't factor nicely.
  • The discriminant b² − 4ac tells you the number of real solutions: positive → 2, zero → 1, negative → 0 real solutions.
Worked example

Solve 2x² + 3x − 2 = 0 using the quadratic formula.

Solution. a=2, b=3, c=−2. Discriminant: 3² − 4(2)(−2) = 9 + 16 = 25. x = (−3 ± 5)/4, giving x = 0.5 or x = −2.

Completing the Square

  • To complete the square on x² + bx, add (b/2)² to make a perfect square trinomial.
  • Used to solve quadratics, rewrite in vertex form, and derive the quadratic formula itself.
Worked example

Solve x² + 6x − 7 = 0 by completing the square.

Solution. Move the constant: x² + 6x = 7. Add (6/2)² = 9 to both sides: x² + 6x + 9 = 16. Factor: (x + 3)² = 16. Take the square root: x + 3 = ±4, so x = 1 or x = −7.

Check yourself

2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.

1. For y = 2x² − 8x + 3, find the axis of symmetry.

2. Solve by factoring: x² + 2x − 15 = 0.

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