Algebra visual reference

Quadratic Formula Guide

The quadratic formula solves every equation in standard form ax² + bx + c = 0 when a is not zero. Keeping the entire numerator over 2a prevents the most common substitution error.

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Quadratic formula guide labeling coefficients a b and c explaining the discriminant and showing a worked quadratic equation example
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How to read it

Use the chart as a decision tool.

A printable quadratic formula guide explaining coefficients, the discriminant, solution types, and a complete worked example. Read the label first, identify the matching structure in your problem, then confirm that any stated conditions apply before substituting values.

01

The formula

For ax² + bx + c = 0 with a ≠ 0.

Solutions

x=−b±b2−4ac2a\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Discriminant

Δ=b2−4ac\displaystyle \Delta=b^2-4ac
02

What the discriminant says

Δ (capital delta) is the discriminant b² − 4ac. Its sign classifies the roots before you finish.

If Δ > 0

Δ>0⇒two distinct real roots\displaystyle \Delta>0\Rightarrow\text{two distinct real roots}

If Δ = 0

Δ=0⇒one repeated real root\displaystyle \Delta=0\Rightarrow\text{one repeated real root}

If Δ < 0

Δ<0⇒two complex roots\displaystyle \Delta<0\Rightarrow\text{two complex roots}
03

Worked example

Solve 2x² − 3x − 2 = 0.

Coefficients

a=2, b=−3, c=−2\displaystyle a=2,\ b=-3,\ c=-2

Substitute

x=3±254\displaystyle x=\frac{3\pm\sqrt{25}}4

Solutions

x=2orx=−12\displaystyle x=2\quad\text{or}\quad x=-\frac12
How to remember it

Write standard form before substituting

  1. 1Move every term to one side so the other side is zero.
  2. 2Copy a, b, and c with their signs.
  3. 3Put −b ± √(b²−4ac) together over the full denominator 2a.
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