Quadratic Equations and Functions

Definition: A quadratic function is a polynomial function of degree 2, written in standard form as f(x) = ax² + bx + c (with a ≠ 0), whose graph is always a symmetric U-shaped curve called a parabola.

How It Works

The coefficient a controls how wide or narrow the parabola is, and its sign controls which way it opens: positive a opens upward (a minimum point), negative a opens downward (a maximum point).
The vertex is the parabola’s turning point, its minimum or maximum, located at x = -b/(2a); substituting that x back into f(x) gives the vertex’s y-coordinate.
The axis of symmetry is the vertical line x = -b/(2a) passing through the vertex. The parabola is a mirror image of itself across this line.
The roots (or zeros, or x-intercepts) are the x-values where f(x) = 0, found using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).
The expression under the square root, b² - 4ac, is called the discriminant. Its sign alone tells you how many real roots the equation has, without needing to finish solving.
A positive discriminant means two distinct real roots (the parabola crosses the x-axis twice); zero means exactly one repeated real root (the parabola’s vertex touches the x-axis); negative means no real roots at all (the parabola never crosses the x-axis).
Factoring solves a quadratic by rewriting it as a product of two linear terms, (x - r1)(x - r2) = 0, then using the fact that a product is zero only if one of its factors is zero. This only works cleanly when the roots are rational.
Completing the square rewrites ax² + bx + c into the form a(x - h)² + k, directly exposing the vertex (h, k) without needing calculus or the quadratic formula.
Quadratics model any situation with constant acceleration or a single interior maximum/minimum: projectile height over time, the shape of a satellite dish, revenue as a function of price.
The quadratic formula itself is simply completing the square done once, in general, on ax² + bx + c = 0, then solved for x, which is why it always works for every quadratic.

Illustration

y = 1.0x² + 0.0x + 0.0 Vertex: (0.0, 0.0) Discriminant: 0.0 Roots: x = 0.0 (repeated)

Drag the sliders below to change a, b, and c. The parabola, its vertex (gold dot), and its real roots (red dots on the x-axis, when they exist) redraw live.

a 1.00 b 0.0 c 0.0

var aIn = document.getElementById(‘quad-a’); var bIn = document.getElementById(‘quad-b’); var cIn = document.getElementById(‘quad-c’); var aOut = document.getElementById(‘quad-a-out’); var bOut = document.getElementById(‘quad-b-out’); var cOut = document.getElementById(‘quad-c-out’); var resetBtn = document.getElementById(‘quad-reset’);

function fmt(n) { return (Math.round(n * 100) / 100).toFixed(2).replace(/.00$/, ‘.0’); }

function update() { var a = parseFloat(aIn.value); var b = parseFloat(bIn.value); var c = parseFloat(cIn.value); aOut.textContent = fmt(a); bOut.textContent = fmt(b); cOut.textContent = fmt(c);

// Sample the curve across the visible x range, clamping y so a steep
// parabola still draws a sensible path instead of shooting far off-chart.
var pts = [];
for (var x = -10; x <= 10.001; x += 0.25) {
  var y = a * x * x + b * x + c;
  if (y > 12) y = 12;
  if (y < -12) y = -12;
  var p = toPx(x, y);
  pts.push((pts.length === 0 ? 'M' : 'L') + p[0].toFixed(1) + ',' + p[1].toFixed(1));
}
curve.setAttribute('d', pts.join(' '));

if (Math.abs(a) < 0.0001) {
  // Degenerates to a line; keep the widget from dividing by zero.
  vertexDot.setAttribute('visibility', 'hidden');
  axisLine.setAttribute('visibility', 'hidden');
  root1.setAttribute('visibility', 'hidden');
  root2.setAttribute('visibility', 'hidden');
  eqTxt.textContent = 'y = ' + fmt(b) + 'x + ' + fmt(c) + ' (a = 0, not quadratic)';
  vertexTxt.textContent = 'Vertex: none (linear)';
  discTxt.textContent = 'Discriminant: n/a';
  rootsTxt.textContent = b !== 0 ? ('Root: x = ' + fmt(-c / b)) : 'No unique root';
  return;
}

vertexDot.setAttribute('visibility', 'visible');
axisLine.setAttribute('visibility', 'visible');

var vx = -b / (2 * a);
var vy = a * vx * vx + b * vx + c;
var vpx = toPx(vx, Math.max(-12, Math.min(12, vy)));
vertexDot.setAttribute('cx', vpx[0]);
vertexDot.setAttribute('cy', vpx[1]);
var axpx1 = toPx(vx, 12), axpx2 = toPx(vx, -12);
axisLine.setAttribute('x1', axpx1[0]); axisLine.setAttribute('y1', axpx1[1]);
axisLine.setAttribute('x2', axpx2[0]); axisLine.setAttribute('y2', axpx2[1]);

var disc = b * b - 4 * a * c;
eqTxt.textContent = 'y = ' + fmt(a) + 'x² + ' + fmt(b) + 'x + ' + fmt(c);
vertexTxt.textContent = 'Vertex: (' + fmt(vx) + ', ' + fmt(vy) + ')';
discTxt.textContent = 'Discriminant: ' + fmt(disc);

if (disc > 0.0001) {
  var r1 = (-b + Math.sqrt(disc)) / (2 * a);
  var r2 = (-b - Math.sqrt(disc)) / (2 * a);
  rootsTxt.textContent = 'Roots: x = ' + fmt(r1) + ', x = ' + fmt(r2);
  var p1 = toPx(r1, 0), p2 = toPx(r2, 0);
  root1.setAttribute('cx', p1[0]); root1.setAttribute('cy', p1[1]); root1.setAttribute('visibility', 'visible');
  root2.setAttribute('cx', p2[0]); root2.setAttribute('cy', p2[1]); root2.setAttribute('visibility', 'visible');
} else if (disc > -0.0001) {
  rootsTxt.textContent = 'Roots: x = ' + fmt(vx) + ' (repeated)';
  var p = toPx(vx, 0);
  root1.setAttribute('cx', p[0]); root1.setAttribute('cy', p[1]); root1.setAttribute('visibility', 'visible');
  root2.setAttribute('visibility', 'hidden');
} else {
  rootsTxt.textContent = 'Roots: none real (discriminant < 0)';
  root1.setAttribute('visibility', 'hidden');
  root2.setAttribute('visibility', 'hidden');
}

}

[aIn, bIn, cIn].forEach(function (el) { el.addEventListener(‘input’, update); }); resetBtn.addEventListener(‘click’, function () { aIn.value = 1; bIn.value = 0; cIn.value = 0; update(); });

update(); })();

Under the Hood

Deriving the quadratic formula by completing the square, starting from ax² + bx + c = 0:

ax² + bx + c = 0
x² + (b/a)x = -c/a                      divide by a
x² + (b/a)x + (b/2a)² = (b/2a)² - c/a    add (b/2a)² to both sides to complete the square
(x + b/2a)² = (b² - 4ac) / 4a²          factor the left side, combine the right
x + b/2a = ±√(b² - 4ac) / 2a            take the square root of both sides
x = (-b ± √(b² - 4ac)) / 2a             solve for x

Every step is reversible and works for any a, b, c with a ≠ 0, which is exactly why the resulting formula is universal.

Worked Example 1: Two real roots. Given: x² - 5x + 6 = 0 (a=1, b=-5, c=6). Step 1: discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1. Step 2: x = (5 ± √1) / 2 = (5 ± 1) / 2. Answer: x = 3 or x = 2 (confirm by factoring: (x-3)(x-2) = 0).

Worked Example 2: Repeated root. Given: x² - 6x + 9 = 0 (a=1, b=-6, c=9). Step 1: discriminant = 36 - 36 = 0. Step 2: x = 6 / 2 = 3. Answer: x = 3 (a double root; the parabola’s vertex sits exactly on the x-axis).

Worked Example 3: No real roots. Given: x² + 2x + 5 = 0 (a=1, b=2, c=5). Step 1: discriminant = 4 - 20 = -16. Answer: no real solutions; the parabola stays entirely above the x-axis (since a > 0 and the vertex’s y-value is positive).

History

Babylonian mathematicians were solving specific quadratic-style problems with geometric methods as early as 2000 BCE, though without any symbolic algebra.
The Persian mathematician Al-Khwarizmi systematized general methods for solving quadratics around 820 CE in his book “Al-Jabr,” the work that gave algebra its name.
Early solutions were entirely geometric (literally completing a physical square), since negative numbers were not yet widely accepted, which is where the technique’s name “completing the square” comes from.
Negative and complex roots were only gradually accepted starting in the 16th and 17th centuries, as symbolic algebra matured through mathematicians like Descartes and Euler.
The compact “x = (-b ± √(b²-4ac))/2a” notation used today is a relatively modern convenience, the underlying method is over a thousand years older than the symbol-heavy formula.

Why It Matters

Projectile motion in physics is quadratic: height over time under constant gravity is exactly a parabola, so the quadratic formula directly gives when a thrown object lands.
Businesses use quadratic revenue models (price × quantity, where raising price lowers quantity sold) to find the exact price that maximizes revenue, at the vertex.
Engineers shape satellite dishes, telescope mirrors, and headlight reflectors as parabolas specifically because of a reflective property: every ray parallel to the axis reflects through a single focus point.
The discriminant’s sign is a fast diagnostic used constantly in engineering and physics to check whether a system has real, physically meaningful solutions before doing any further work.
Optimization problems across economics, biology, and computer science frequently reduce to finding a quadratic function’s vertex, its single guaranteed maximum or minimum.

Common Pitfalls

Forgetting the ± in the quadratic formula, and reporting only one root when two exist.
Assuming every quadratic factors into nice integers. Most real-world quadratics need the formula or completing the square instead.
Mixing up the vertex formula’s sign: it is x = -b/(2a), not b/(2a); dropping the negative sign silently flips the parabola’s turning point to the wrong side.
Confusing “no real roots” with “no solution at all.” A negative discriminant means no real roots, but the equation still has two complex roots.
Misreading a downward-opening parabola’s vertex as a minimum. When a is negative, the vertex is the maximum, not the minimum.

Comparison

Method	Best When	Gives Directly
Factoring	Roots are rational, equation factors cleanly	Roots
Completing the square	You need the vertex form	Vertex (h, k)
Quadratic formula	Always works, any a/b/c	Roots (real or complex)
Graphing	Visual estimate, checking work	Approximate roots and vertex

FAQ

Why does a negative discriminant mean no real roots? The quadratic formula requires taking the square root of the discriminant. A negative number has no real square root, only an imaginary one, so the formula only produces real answers when the discriminant is zero or positive.

Is there a “cubic formula” for degree-3 equations the same way? Yes, though it is far more complicated, involving nested cube roots and complex intermediate steps even when the final answer is a real number. Formulas exist through degree 4; Abel and Galois proved in the 1820s-30s that no general algebraic formula exists for degree 5 and higher.

Example

A ball thrown upward at 20 m/s from a 5 m platform has height h(t) = -4.9t² + 20t + 5. Setting h(t) = 0 and applying the quadratic formula gives the time it lands, about 4.3 seconds, directly from the same formula used above.

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