Quadratic Formula Calculator and Quadratic Equation Solver — Step-by-Step
Paste an equation or enter a, b, and c for ax² + bx + c = 0 to get exact roots, rounded decimals, the discriminant, a graph, and full working.
Enter a quadratic
Accepts fractions, decimals, √ radicals, parentheses, implicit multiplication, and terms on either side. Use sqrt(2) or √2. Without =, the expression is set equal to zero.
Quick problem cases
Keyboard: arrow keys change input tabs; Ctrl/Cmd + Enter solves.
Solution
Enter an equation and choose Solve.
Your exact and decimal roots will appear here.
Step-by-step working
Steps will appear here.
Properties and alternate forms
Derived properties
Vertex
Awaiting solution
Axis of symmetry
Awaiting solution
Opening
Awaiting solution
Extremum
Awaiting solution
y-intercept
Awaiting solution
Alternate forms
Standard form
Awaiting solution
Vertex form
Awaiting solution
Factored form
Awaiting solution
Graph and intercepts
Curvex-interceptsVertexy-intercept
The graph will show the parabola, intercepts, vertex, and axis of symmetry once solved.
Calculator methodology and review
Developed and reviewed by
Starlight Robotics calculator team
Last reviewed
Formulas
Δ = b² − 4ac; x = (−b ± √Δ)/(2a); Vieta checks
Input limits
One variable x, polynomial degree ≤ 2, up to 500 characters; real numeric coefficients
Exactness and rounding
Integers, fractions, and finite decimals use rational arithmetic. Simplified exact roots are separated from approximations rounded to 10 places.
A quadratic equation is any equation that can be written in the form
ax² + bx + c = 0 where a, b, and c are real numbers and
a ≠ 0. Quadratics appear everywhere: projectile motion, optimization
problems, area/geometry questions, and many modeling tasks in science and engineering.
The Quadratic Formula
Every quadratic can be solved using the quadratic formula:
x = [-b ± √(b2 - 4ac)] / (2a)
The expression under the square root, b² − 4ac, is called the
discriminant (Δ). It determines the nature of the roots:
If Δ > 0 → two distinct real roots.
If Δ = 0 → one real repeated root.
If Δ < 0 → two complex conjugate roots.
How to use this calculator
Paste an equation as written, or switch to Coefficients and enter a, b, and c.
Choose the quadratic formula, factoring, or completing the square for the working.
Click Solve Equation to compute the discriminant and roots.
Review the exact answer first, then decimals, verification, properties, and graph.
Everything runs client-side in your browser, so your inputs never leave your device.
Identify a, b, and c
First move every term to one side. In x² + 4 = 5x, standard form is x² − 5x + 4 = 0, so a=1, b=−5, c=4. A missing term has coefficient zero: 2x²−7=0 has b=0.
Choose a solving method
Factor when simple factor pairs are visible. Complete the square to expose vertex form or derive the formula. Use the quadratic formula when factors are unclear, or roots are irrational or complex.
Read the graph and discriminant
Δ > 0 gives two x-intercepts, Δ = 0 gives one tangent intercept, and Δ < 0 gives none. The vertex is at x=−b/(2a); the sign of a determines whether the parabola opens up or down.
Check an answer
Substitute a root into both sides of the original equation. You can also check both roots with Vieta: r₁+r₂=−b/a and r₁r₂=c/a.
Fully worked examples for every root type
Two distinct real roots
x²−5x+6=0: Δ = 25−24 = 1. Thus x=(5±1)/2, so x=3 or x=2.
Vieta check: 3+2=5=−b/a and 3·2=6=c/a.
One repeated root
x²−6x+9=0: Δ = 36−36 = 0. Thus x=6/2=3, repeated.
Substitution: 3²−6·3+9=9−18+9=0.
Irrational radical roots
2x²−8x+5=0: Δ = 64−40 = 24. Then x=(8±√24)/4=(4±√6)/2.
Vieta check: the sum is 4=−b/a; the product is (16−6)/4=5/2=c/a.
Complex conjugate roots
x²+4x+8=0: Δ = 16−32 = −16. Then x=(−4±4i)/2=−2±2i.
Vieta check: the sum is −4; the product (−2+2i)(−2−2i)=4+4=8.
Fractional coefficients
(3/4)x²−(5/2)x+2=0. Multiply by 4: 3x²−10x+8=0. Δ = 100−96 = 4, so x=(10±2)/6: x=2 or x=4/3.
Vieta check: 2+4/3=10/3=−b/a and 2·4/3=8/3=c/a.
Practical input and interpretation guide
1
Move terms automatically
You may enter terms on either side, such as x²+4=5x. The solver subtracts the right side and shows the resulting standard form.
Equation input
2
Use zero for missing terms
For x²−9=0, enter a=1, b=0, c=−9 in coefficient mode—or simply omit the x term in equation mode.
Coefficients
3
Exact before approximate
(4+√6)/2 is exact. Its decimal is rounded, so use the exact form in symbolic work and the decimal for measurement or plotting.
Result reading
4
Roots and x-intercepts
Real roots are x-intercepts because they make the polynomial’s y-value zero. Complex roots do not appear as intercepts on a real-coordinate graph.
Graph meaning
5
Linear fallback
If a=0, the solver identifies the input as linear and solves bx+c=0 instead of applying a quadratic formula that would divide by zero.
Edge case
Quadratic Equation: FAQs
What is the quadratic formula?
x = [−b ± √(b² − 4ac)]/(2a). It solves ax²+bx+c=0 for x when a ≠ 0.
What does the discriminant (Δ) tell me?
Δ=b²−4ac. If Δ>0 there are two real roots; if Δ=0 there is one repeated real root; if Δ<0 there are two complex roots.
Are fractions and decimals allowed?
Yes. Use forms such as 3/4, -0.5, or (1/2)x². Finite decimals are kept as exact fractions for algebra; radicals in coefficients are accepted and evaluated.
What happens when a = 0?
The expression is not quadratic. The calculator solves the remaining linear equation bx+c=0, or reports an identity or contradiction when b is also zero.
How do I enter a missing b or c term?
Omit the term in equation mode, or enter 0 for its coefficient. For x²+5=0, b=0; for x²−3x=0, c=0.
Why can a quadratic have no real roots?
When b²−4ac is negative, the graph does not cross the x-axis and the two solutions are complex conjugates.
How are exact and decimal roots different?
An exact answer preserves fractions and simplified radicals. A decimal answer is a rounded approximation, shown to up to 10 decimal places.
When should I factor instead of using the quadratic formula?
Factoring is usually fastest when integer or simple rational factors are visible. The quadratic formula always works and is preferable for irrational or complex roots.
How can I verify the roots?
Substitute each root into the original equation, or check Vieta’s formulas: the sum is −b/a and the product is c/a.
Why are roots the x-intercepts?
A root makes y=ax²+bx+c equal zero. On the graph, y=0 is the x-axis, so real roots occur exactly where the parabola meets that axis.
