Rational root theorem first
AskSia lists all possible rational roots from p over q (factors of constant term over factors of leading coefficient) and tests each by substitution until one is found.
Rational root
AI Solver by AskSia
Cubic Function Solver: three roots, always
Type or photograph any cubic ax^3 + bx^2 + cx + d. AskSia applies the rational root theorem to find one rational root, divides out by synthetic division to reduce to a quadratic, applies the quadratic formula for the other two roots, and graphs the cubic with all three roots labeled.
Works with word problems, equations, code, and science prompts.
∫ 3x² · sin(x) dx
SubjectsCalculusAlgebraPhysicsChemistryBiologyCSStatisticsEcon
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asksia.ai/solver
98% · Verified
Problem
Find the area enclosed between y = x² and y = 2x in the first quadrant.
1
Find the points of intersection
Set the curves equal: x² = 2x → x(x - 2) = 0, so x = 0 and x = 2.
2
Identify which curve is on top
On (0, 2), 2x > x², so y = 2x is the upper curve.
3
Set up the area integral
A = ∫₀² (2x - x²) dx
4
Integrate term-by-term
A = [ x² - x³/3 ]₀²
Final Answer
A = 4/3 sq. units
Quick Answer
What is the AskSia Cubic Function Solver?
AskSia's Cubic Function Solver is an AI tool that finds all three roots (real or complex) of any cubic function ax^3 + bx^2 + cx + d. AskSia tries factoring first, then applies the rational root theorem to find one rational root, divides out by synthetic division to leave a quadratic, and uses the quadratic formula for the remaining two roots. The fundamental theorem of algebra guarantees exactly three roots counting multiplicity, and AskSia returns all of them with substitution check and graph.
98%
solution accuracy
50M+
problems solved
~1.5s
avg solve time
A+
study-ready explanations
Why AskSia Solver
Three roots. Every cubic.
Rational root, synthetic division, quadratic formula. AskSia chains the methods together to return all three roots with the work shown.
Synthetic division shown
Once a rational root is found, AskSia divides the cubic by (x minus root) using synthetic division. The quotient is a quadratic, with the work shown clearly.
Synthetic division
Quadratic formula on the depressed quadratic
The quotient quadratic is solved by the formula. AskSia computes the discriminant, classifies the roots (two real, one repeated, two complex), and returns the values.
Quadratic formula
Complete factorization
AskSia writes the cubic as a product of linear factors over the reals or the complex numbers. The factorization is verified by expanding back to the original.
Factored form
Graph with turning points
Auto-generated graph of the cubic with real roots marked, turning points (where the derivative is zero) labeled, and end behavior shown. The visual confirms the shape and root count.
Graph included
Substitution check on each root
Every root is substituted into the original cubic to confirm it evaluates to zero. The check is shown for each root.
Verified
How It Works
Three taps to all three roots.
Step 01
Capture the cubic
Snap a photo of the cubic from your textbook, paste from a homework portal, or type with LaTeX support.
Input mode
Snap a Photo
Textbook, handwriting, screenshot
Paste Text
Word problem or equation
Calculator
LaTeX-ready equation editor
Step 02
Watch Sia work the methods
Rational root theorem applied, synthetic division performed, then the quadratic formula on the quotient. Each step has a plain-English reason.
1
Set curves equal
x² = 2x → x = 0, x = 2
2
Set up the integral
A = ∫₀² (2x - x²) dx
3
Evaluate
A = [x² - x³/3]₀² = 4/3
Step 03
See all roots and the graph
Three roots returned with multiplicities, complete factorization, graph with real roots and turning points labeled, and substitution check on each root.
Auto-generated diagram
Region between y = 2x and y = x² — area = 4/3
Available On
Solve anywhere
you study.
Every solve syncs across Web, iOS, and Android — start it at your desk, finish on your phone.
Web App
Full study studio
Split-panel interface with the worked solution on the left, the auto-generated diagram and AI tutor chat on the right.
Drag & drop image upload + LaTeX equation editor
Auto-generated diagrams render alongside steps
Side-panel AI tutor chat for hints and alt methods
Export to PDF, DOCX, Notion, or Google Docs
Hi! What are we studying today?
Ask about your homework, lecture, or readings...↑
Calculus
98% verified
1.4s
Step 4 of 4 · Evaluate
A = [x² - x³/3]₀² = 4/3
Mobile App
Snap & solve, anywhere
Open the camera, frame the problem, and the worked solution plus diagram appear in seconds.
One-tap snap-and-solve on iOS and Android
Pinch-to-zoom diagrams, swipe between steps
Auto-sync solves with your Web library
Offline review of saved solutions and flashcards
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Calc
98%
1.4s
Area between y=2x & y=x²
A = 4/3 sq. units ✓
Use Cases
Every cubic, every chapter.
Algebra 2 cubic homework
Solve ax^3 + bx^2 + cx + d = 0 from Algebra 2. AskSia walks through rational root and synthetic division step by step.
Algebra 2
Pre-Calc and College Algebra
Cubic functions with complex roots. AskSia returns conjugate pairs and writes the full factorization over the complex numbers.
Pre-Calc
Factor a cubic completely
When the prompt asks to factor rather than solve, AskSia returns the complete factorization. Over the reals, irreducible quadratics are left in place.
Factorization
Graph a cubic and identify features
Find x-intercepts (roots), turning points (local max and min), and end behavior. AskSia labels all on the graph.
Graphing
Word problems with cubics
Volume problems, polynomial models from physics or economics. AskSia builds the cubic from the prose and solves.
Word problems
Check your roots
Paste your candidate roots and AskSia substitutes each back into the original cubic, flagging any that fail.
Root check
Compare
AskSia vs. ChatGPT,
Photomath & Symbolab.
General chatbots hallucinate. Photo solvers stop at math. AskSia is built for actual coursework with verified accuracy, visual learning, and every subject.
| Feature | AskSia Solver | ChatGPT | Photo Solvers |
|---|---|---|---|
| Solution accuracy | ✓ 98% | ~70-85%, hallucinations | ~90%, math only |
| Auto-generated diagrams | ✓ Every solve | Inconsistent / broken | Graphs only, math-only |
| Step-by-step explanations | ✓ Numbered + plain English | Inconsistent depth | ✓ Math steps |
| Subject coverage | ✓ Math, Physics, Chem, Bio, CS, Econ | ✓ Wide but unverified | Math only |
| Photo input | ✓ Handwriting + diagrams + code | Photos OK, weak on handwriting | ✓ Math photos only |
| Answer verification | ✓ Self-checked before display | No verification | Math engine only |
| Tutor follow-ups | ✓ Hints, alt methods, ELI5 | ✓ General chat | Not available |
| Practice and flashcards | ✓ One-tap from any solve | Manual prompting | Not available |
| Code debugging | ✓ Python, Java, C++, SQL... | ✓ Yes | Not available |
| Free to start | ✓ Daily solves, no card | Limited model access | Steps locked behind paywall |
FAQ
Frequently asked questions.
How does AskSia find roots of a cubic?
First, AskSia tries factoring by grouping if the coefficients allow. If not, the rational root theorem lists all possible rational roots from p over q, where p divides the constant term and q divides the leading coefficient. AskSia tests each candidate by substitution. Once one root is found, synthetic division reduces the cubic to a quadratic, which is solved by the quadratic formula. The result is all three roots of the cubic.
Why does a cubic always have three roots?
The fundamental theorem of algebra says a polynomial of degree n has exactly n roots, counting multiplicity. For a cubic (degree 3), that means three roots. The roots may be three distinct real numbers, one real and a complex conjugate pair, or a repeated real root with another root (a double root and a single root). AskSia returns all three with multiplicities labeled.
Can AskSia handle cubics with no rational roots?
Yes. When the rational root theorem returns no roots, AskSia falls back to numeric methods (the cubic formula or Cardano's method for an exact answer, or numeric root-finding for a decimal approximation). The method used is named in the response, and the roots are returned with high precision.
Does AskSia show the graph of the cubic?
Yes. Every cubic solve comes with an auto-generated graph showing the cubic curve, the real roots marked on the x-axis, the turning points (local maximum and minimum) labeled, and the end behavior indicated. The graph confirms the number of real roots visually and helps with understanding the cubic's shape.
How accurate is AskSia?
AskSia is engineered for accuracy on standard high school and college coursework. Accuracy comes from subject-specialized models, a symbolic verification pass that catches arithmetic errors, and a self-check step that re-derives the answer before showing it to you.
Can I get practice problems and flashcards?
Yes. After any solve, ask Sia to generate similar practice problems at SAT, ACT, AP, IB, or college difficulty, or build a flashcard set on the underlying concept in one tap. Useful for exam prep and spaced repetition before a quiz, midterm, or final.
How much does AskSia cost?
AskSia has a free plan that includes daily solves across all subjects. AskSia Pro and Super include unlimited solves, advanced subjects, the full AI tutor companion, exports, and priority response speed. See pricing for details.
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Every cubic. Three roots. The full factorization.
Join 2M+ students using AskSia to find all real and complex roots of any cubic, with the rational root theorem, synthetic division, and the quadratic formula chained together and the graph drawn for every solve.
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