Mathematics.

polynomial equations

Quadratic Equation

Algebra I30 minDifficulty2 out of 10

You should know: natural numbers

Overview

A quadratic equation is a polynomial equation of degree 2 — the highest power of the variable is 2. Its graph is a parabola, and solving it means finding where that parabola crosses the x-axis. The quadratic formula gives a universal method for solving any quadratic equation, no matter how ugly its coefficients.

Intuition

A quadratic equation ax² + bx + c = 0 asks: at what x-values does the parabola y = ax² + bx + c touch or cross the x-axis (where y=0)? A parabola can cross the x-axis twice, touch it exactly once (a repeated root), or miss it entirely (no real roots) — three qualitatively different pictures, all captured by the sign of a single quantity in the formula: the discriminant.

Formal Definition

Definition

The general quadratic equation and the quadratic formula that solves it:

ax2+bx+c=0,a≠0ax^2 + bx + c = 0,\quad a \neq 0

Standard form

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

Notation

NotationMeaning
a,b,ca, b, cCoefficients: a is the leading (x²) coefficient, b the linear coefficient, c the constant term
Δ=b2−4ac\Delta = b^2-4acThe discriminant, which determines the number and type of roots

Derivation

Deriving the quadratic formula by completing the square:

ax2+bx+c=0  ⟹  x2+bax=−caax^2+bx+c=0 \implies x^2 + \frac{b}{a}x = -\frac{c}{a}

Divide by a, isolate the constant

x2+bax+(b2a)2=−ca+(b2a)2x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2

Add (b/2a)² to both sides to complete the square

(x+b2a)2=b2−4ac4a2\left(x+\frac{b}{2a}\right)^2 = \frac{b^2-4ac}{4a^2}

Left side is now a perfect square

x+b2a=±b2−4ac2a  ⟹  x=−b±b2−4ac2ax + \frac{b}{2a} = \pm\frac{\sqrt{b^2-4ac}}{2a} \implies x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}

Take the square root and solve for x

Properties

Discriminant > 0

Two distinct real roots\text{Two distinct real roots}

Discriminant = 0

Exactly one repeated real root\text{Exactly one repeated real root}

Discriminant < 0

Two complex conjugate roots, no real roots\text{Two complex conjugate roots, no real roots}

Vieta's formulas

x1+x2=−ba,x1x2=cax_1+x_2 = -\frac{b}{a},\quad x_1 x_2 = \frac{c}{a}

Example: Sum and product of roots without solving explicitly

Applications

Projectile motion (height as a function of time under gravity) is modeled by a quadratic; solving for when height=0 gives landing time.

Formula Explorer

Adjust a, b, c and watch the parabola and roots update

Loading visualization…

Worked Examples

  1. Factor: look for two numbers multiplying to 6 and adding to -5.

    (x−2)(x−3)=0(x-2)(x-3) = 0

Answer: x = 2 or x = 3

Practice Problems

Difficulty 3/10

Solve x² + 4x + 4 = 0.

Difficulty 5/10

A ball is thrown upward with height h(t) = −5t² + 20t + 2 (metres). When does it hit the ground?

Difficulty 5/10

A rectangular garden is 3 m longer than it is wide and has area 40 m². Find its dimensions.

Common Mistakes

Common Mistake

Forgetting the ± in the quadratic formula and reporting only one solution.

A quadratic generically has TWO roots; always check both +√Δ and -√Δ unless the discriminant is exactly zero.

Common Mistake

Dividing by a coefficient that could be zero without checking a ≠ 0 first.

If a=0, the equation is linear, not quadratic, and the quadratic formula (which divides by 2a) is undefined.

Quiz

If the discriminant of a quadratic is negative, how many real roots does it have?

Flashcards

1 / 2

Historical Background

Babylonian mathematicians solved specific quadratic equations using geometric methods as early as 2000 BCE, though without algebraic notation. The Persian mathematician Al-Khwarizmi systematized methods for solving quadratics in his 9th-century treatise (the source of the word 'algebra', from 'al-jabr'). The modern symbolic quadratic formula, using the ± notation we recognize today, was popularized in Europe by the 16th and 17th centuries as algebraic notation matured.

  1. c. 2000 BCE

    Babylonians solve quadratic-type problems geometrically

  2. c. 820 CE

    Al-Khwarizmi systematizes quadratic-solving methods in Al-Jabr

    Muhammad ibn Musa al-Khwarizmi

  3. 16th–17th century

    Modern symbolic quadratic formula notation solidifies in Europe

Summary

  • A quadratic equation ax²+bx+c=0 (a≠0) is solved by the quadratic formula x=(-b±√(b²-4ac))/2a.
  • The discriminant Δ=b²-4ac determines the number/type of roots: positive→2 real, zero→1 repeated, negative→2 complex.
  • Derived by completing the square on the general form.
  • Vieta's formulas relate the roots directly to the coefficients without solving explicitly.

References

  1. WebsiteWikipedia — Quadratic equation