Quadratic Equation Formulas
Learn the essential quadratic equation formulas for solving, checking roots, and graphing a parabola, with one connected example.
These six formulas describe the same quadratic from different angles: its coefficients, roots, discriminant, vertex, graph, and root relationships.
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6 formulas, different jobs
These six formulas describe the same quadratic from different angles: its coefficients, roots, discriminant, vertex, graph, and root relationships.
Standard Form
Use it to identify a, b, and c before applying another formula.
Quadratic Formula
Use it to find every real or complex solution.
Discriminant
Use it to predict the number and type of roots.
Vertex Formula
Use it to find the x-coordinate of the vertex or axis of symmetry.
Vertex Form
Use it to read the vertex and graph the parabola.
Vieta’s Formulas
Use them to check the sum and product of the roots.
One equation, four useful results
For 2x² − 5x − 3 = 0, move from the coefficients to the roots, then check and graph the answer.
- 1 · Identify
- 2 · Predict
- 3 · Solve
- 4 · Graph
Common mistakes
- Losing the sign of b when substituting into the quadratic formula.
- Dividing only the square-root term instead of the entire numerator by 2a.
- Using the vertex formula as if it returns both roots.
See which formulas your problem uses
Enter a problem and get a step-by-step solution with the formulas and theorems identified.
Calculators and related formula guides
Equation Calculator
Solve a quadratic equation and see each step.
Algebra Calculator
Work through factoring, roots, and algebraic forms.
Slope formulas
Compare the slope formula, point-slope form, slope-intercept form, standard form, and special horizontal and vertical lines.
Exponent rules
Learn the product, quotient, power, zero, negative, and fractional exponent rules with examples and one complete simplification.
Quadratic equations FAQ
Which quadratic formula should I use first?
Write the equation in standard form first. Then use the discriminant to inspect the roots or the quadratic formula to solve directly.
Are vertex form and the vertex formula the same?
No. The vertex formula finds the x-coordinate of the vertex, while vertex form displays the complete vertex as (h, k).