Quartic Equation Calculator
A quartic equation is a polynomial equation of the fourth degree, written ax^4 plus bx^3 plus cx^2 plus dx plus e equals zero, where a is not zero. It is the highest-degree polynomial whose roots can always be found by an exact algebraic formula, although in practice the formula is long, so this calculator finds the roots numerically to high precision instead. Enter the five coefficients and the calculator returns the real roots, the values of x that make the left side equal zero. It works by scanning for sign changes of the polynomial and refining each crossing with the bisection method, a reliable technique that brackets a root and halves the interval until it is pinned down. A quartic can have zero, two or four real roots, and any roots that are complex are not shown because this tool reports real solutions only. The five coefficients are fully editable so you can solve any fourth-degree equation, from a textbook exercise to a characteristic equation in engineering or physics. Every figure here is computed deterministically from the coefficients you enter, never estimated or guessed, so the same input always produces the same roots. The method and a worked example that reconciles to the calculator default are shown in full below.
A quartic solves ax^4 + bx^3 + cx^2 + dx + e = 0 for x. With the default coefficients 1, -10, 35, -50 and 24, the four real roots are 1, 2, 3 and 4, because the polynomial factors as (x-1)(x-2)(x-3)(x-4).
Formula
a x^4 + b x^3 + c x^2 + d x + e = 0
Roots are the values of x where the polynomial equals zero
Found by sign-change scanning and bisection to high precision
The calculator evaluates the polynomial across a range, detects where it changes sign (each sign change brackets a real root), and narrows each bracket with repeated bisection until the root is located to within a very small tolerance.
Worked example
Solve x^4 minus 10x^3 plus 35x^2 minus 50x plus 24 equals 0, using coefficients 1, -10, 35, -50 and 24.
- The polynomial factors as (x-1)(x-2)(x-3)(x-4)
- Each factor is zero at x = 1, x = 2, x = 3 and x = 4
- Check x = 2: 16 - 80 + 140 - 100 + 24 = 0, confirming a root
- The four real roots are 1, 2, 3 and 4
These are the calculator's default inputs, so the roots above match the widget exactly.
Quartic Equation Calculator: frequently asked questions
How many roots does a quartic have?
A fourth-degree polynomial has exactly four roots counted with multiplicity, but some may be complex. Over the real numbers it can have zero, two or four real roots. This calculator reports the real roots.
What method does this use?
It scans the polynomial for sign changes, since the value must pass through zero wherever it changes sign, then refines each bracket with the bisection method until the root is located to high precision.
Why does it not show complex roots?
Complex roots come in conjugate pairs and require a different presentation. This tool focuses on real roots, which are the solutions you read off the x-axis of the graph.
Can a equal zero?
No. If a is zero the equation is not a quartic but a cubic or lower. Keep the leading coefficient a non-zero for a true fourth-degree equation.
Are the roots exact?
They are computed to a very small numerical tolerance, accurate to the decimal places shown. For nice integer roots like the default, the result is exact.
Official sources
- Polynomial root-finding and numerical methods reference: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.