Solve ax² + bx + c = 0 instantly with full step-by-step working. Shows discriminant, real roots, complex roots, and the complete solution method.
enter a, b, c values
a (x² coefficient)
b (x coefficient)
c (constant)
Result (click to copy)
Reading Your Roots
Two real roots means the parabola crosses the x-axis twice
Graphically, this is the most common case for a parabola that opens upward from below zero or downward from above zero — the two x-values are where the curve crosses y=0.
A repeated root means the parabola just touches the x-axis
The vertex of the parabola sits exactly on the x-axis — there's only one x-intercept, counted with "multiplicity 2" in algebra.
Complex roots mean the parabola never touches the x-axis
This isn't "no solution" — the equation still has two solutions, they're just not real numbers. The parabola stays entirely above or entirely below the x-axis.
How to Use the Quadratic Equation Solver
Enter coefficient a — the number multiplying x². Must not be zero (if a=0, the equation is linear, not quadratic).
Enter coefficient b — the number multiplying x. Can be zero.
Enter coefficient c — the constant term. Can be zero.
Click Solve — the calculator shows the discriminant, roots, and full step-by-step working.
Interpret the result — two real roots (discriminant > 0), one repeated root (discriminant = 0), or two complex roots (discriminant < 0).
📐 Quick formula: x = (-b ± √(b²-4ac)) / 2a. The ± gives two solutions: one using + and one using −. The expression b²-4ac under the square root is the discriminant — it determines how many real solutions exist before you do any calculation.
Understanding Quadratic Equations
📐 Standard Form
ax² + bx + c = 0 where a ≠ 0. The highest power of x is 2 — this is what makes it quadratic (from Latin quadratus = square). Every quadratic has exactly 2 solutions in the complex number system, though they may be equal or non-real.
🔢 The Discriminant
b² − 4ac determines the nature of roots. Positive: two distinct real roots. Zero: one repeated real root (the parabola touches the x-axis at exactly one point). Negative: two complex conjugate roots (the parabola never crosses the x-axis). Calculating discriminant first reveals the solution type instantly.
📊 Factoring Method
When ac is small and integer roots exist, factoring is faster: x² − 5x + 6 = (x−2)(x−3) = 0, so x=2 or x=3. Find two numbers that multiply to c and add to b. For x² − 5x + 6: numbers that multiply to 6 and add to −5 are −2 and −3. Factoring only works when integer or simple fraction roots exist.
🌀 Completing the Square
Transforms ax² + bx + c into a(x+h)² + k form. Divide by a, move c to right side, add (b/2a)² to both sides, factor left side. Produces the vertex form of the parabola. The quadratic formula is derived by completing the square on the general form ax² + bx + c = 0.
📈 Parabola Connection
y = ax² + bx + c is a parabola. The roots are where the parabola crosses the x-axis (y=0). Two roots: crosses twice. One root: touches x-axis at vertex. No real roots: parabola entirely above or below x-axis. The vertex is at x = −b/2a, y = c − b²/4a.
🌐 Real Applications
Projectile motion: h = −16t² + v₀t + h₀ (height vs time, find when object hits ground). Area problems: room of area 24m² where length is 2m more than width: w(w+2)=24. Profit maximisation: P = −x² + 40x − 300 (find maximum profit). Engineering stress calculations. Computer graphics curve generation.
Quadratic Equations in Practice
Choosing the right method
For simple equations with small integer coefficients: try factoring first (fastest when it works). For equations where the discriminant is a perfect square: factoring or quadratic formula both work cleanly. For equations with irrational or complex roots: quadratic formula is the only practical method. For understanding the geometry: completing the square gives vertex form directly. The quadratic formula always works regardless of coefficient values — it is the universal fallback.
Complex roots in engineering
Complex roots (when discriminant < 0) appear in electrical engineering (AC circuit analysis), control systems (stability analysis), and signal processing (filter design). Complex roots x = a ± bi indicate oscillatory behaviour in physical systems. The real part (a) represents decay/growth rate; the imaginary part (b) represents oscillation frequency. Engineers regularly work with quadratics that produce complex roots.
Checking your answers
Always verify quadratic solutions by substituting back into the original equation. If x=3 is a root of x² − 5x + 6 = 0: 3² − 5(3) + 6 = 9 − 15 + 6 = 0 ✓. For complex roots: substitute the complex number and verify the result is zero using complex arithmetic. This verification step catches calculation errors and builds confidence in the solution.
📐 Discriminant decision tree: Calculate b²−4ac first. If > 0 → two real roots (use quadratic formula or factor). If = 0 → one root: x = −b/2a (no ± needed). If < 0 → complex roots: real part = −b/2a, imaginary part = √(|discriminant|)/2a. This decision saves calculation time by identifying the solution type upfront.
The Quadratic Formula
x = (−b ± √(b² − 4ac)) ÷ 2a, where the discriminant (b² − 4ac) determines the root type: positive → two real roots, zero → one repeated root, negative → two complex roots.
Linear fallback: if a = 0, the equation isn't quadratic at all — it reduces to bx + c = 0, solved directly as x = −c/b (when b ≠ 0). For example, with a=0, b=2, c=−8: 2x − 8 = 0 → x = 4.
Avoid These Mistakes
Concluding "no solution" for a negative discriminant
A negative discriminant means the roots are complex, not that the equation has no solutions. Every quadratic equation has exactly two roots (real or complex) — a negative discriminant just means they're not real numbers.
Forgetting the ± sign
Dropping the ± and computing only one value misses half the solutions — always compute both x₁ and x₂ unless the discriminant is exactly 0.
Sign errors substituting negative coefficients
When b or c is negative, −b and −4ac need careful sign handling — e.g. for b=−5, −b=5, not −5. Substitute the raw values into the formula and simplify step by step rather than doing the arithmetic in your head.
Treating a=0 as an error instead of a linear equation
If the leading coefficient is 0, the equation is linear, not invalid — it still has a solution (unless b is also 0), just found by simpler algebra than the quadratic formula.
Frequently Asked Questions
What is a quadratic equation?
A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. The highest power of x is 2. They appear in physics (projectile motion), engineering, finance (break-even analysis), and many real-world problems.
What is the discriminant?
The discriminant is b² - 4ac. If positive: two distinct real roots. If zero: one repeated root. If negative: two complex (imaginary) roots. The discriminant tells you how many solutions exist before solving.
What is the quadratic formula?
x = (-b ± √(b² - 4ac)) / 2a. This formula always works for any quadratic equation. The ± means you get two solutions: one with + and one with −.
References
📐 Fundamental Theorem of Algebra
Guarantees every quadratic equation has exactly two roots (counting multiplicity) over the complex numbers — the mathematical basis for why a negative discriminant still yields two valid (complex) solutions rather than none.
📄 Your course textbook or instructor
This tool covers the quadratic formula method. If your coursework requires solving by factoring or completing the square, use this tool to check your answer, not to replace showing that specific method's working.