Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0 with step-by-step discriminant analysis.
Equation
Updates as you typeParabola
Step-by-step solution
Formula
- a
- Coefficient of x² (leading term)
- b
- Coefficient of x (linear term)
- c
- Constant term (also the y-intercept)
- D
- Discriminant = b² − 4ac — the sign of D tells you how many real roots there are
- a, b, c = —
- D = b² − 4ac = —
- √D = —
- 2a = —
- x = —
The sign of the discriminant classifies the roots: positive yields two distinct real roots (the parabola crosses the x-axis twice), zero yields one repeated real root (the parabola touches the x-axis), negative yields a complex-conjugate pair (the parabola does not touch the x-axis).
- (h, k)
- Coordinates of the vertex (h = −b / 2a, k = c − b² / 4a)
- a
- Same leading coefficient as in standard form
Vertex form makes the turning point obvious. When a and k have the same sign, the parabola doesn't cross the x-axis and the roots are complex.
- r₁, r₂
- The two real roots — where the parabola crosses the x-axis
Only works when the discriminant is non-negative (real roots exist). For complex roots, factored form is expressed with complex conjugates r₁ = p + qi, r₂ = p − qi.
- Sum
- Sum of the roots equals −b/a regardless of whether the roots are real or complex
- Product
- Product of the roots equals c/a — a quick sanity check after factoring
Vieta's formulas let you verify factorizations without computing the roots explicitly, and generalize to higher-degree polynomials.
Examples
How It Works
When D > 0, the parabola crosses the x-axis at two distinct points, giving two real roots. When D = 0, the parabola just touches the x-axis at its vertex, producing one repeated root. When D < 0, the parabola never reaches the x-axis, so the solutions are complex conjugates involving the imaginary unit i.
The quadratic formula x = (−b ± √D) / (2a) handles all three cases. This solver computes the discriminant, applies the formula, finds the vertex at (−b/2a, f(−b/2a)), and graphs the parabola so you can visualize the relationship between the equation and its solutions.
Tips & Best Practices
Frequently Asked Questions
What is a quadratic equation?
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. The 'quadratic' name comes from 'quad' meaning square, since the variable is squared.
What does the discriminant tell us?
The discriminant (b² − 4ac) reveals the nature of the roots. If positive, there are two distinct real solutions. If zero, there is exactly one real solution (a repeated root). If negative, there are two complex conjugate solutions.
When are there no real solutions?
A quadratic has no real solutions when the discriminant (b² − 4ac) is negative. Graphically, this means the parabola does not cross the x-axis. The solutions exist as complex numbers involving the imaginary unit i.
What are the different methods for solving quadratics?
You can solve using the quadratic formula, factoring, completing the square, or graphing. The quadratic formula works for all cases, while factoring is quickest when the equation has simple integer roots.
What is the vertex of a parabola?
The vertex is the highest or lowest point on the parabola. For y = ax² + bx + c, the vertex x-coordinate is −b/(2a) and the y-coordinate is found by substituting back. If a > 0 the vertex is a minimum; if a < 0 it is a maximum.
How do you factor a quadratic equation?
To factor ax² + bx + c, find two numbers that multiply to a·c and add to b. Rewrite the middle term using those numbers, then factor by grouping. For example, x² − 5x + 6 factors as (x − 2)(x − 3) because −2 × −3 = 6 and −2 + −3 = −5.
What is completing the square?
Completing the square rewrites ax² + bx + c into the form a(x − h)² + k. Move c to the other side, add (b/2a)² to both sides, then factor the left as a perfect square. This method is how the quadratic formula itself is derived.
Can a quadratic equation have only one solution?
Yes. When the discriminant equals zero, the quadratic has exactly one repeated real root. Graphically, the parabola just touches the x-axis at one point (the vertex). Examples include x² − 6x + 9 = 0, which has the single root x = 3.
What are Vieta's formulas?
Vieta's formulas relate the roots of a quadratic to its coefficients. For ax² + bx + c = 0 with roots r₁ and r₂: the sum r₁ + r₂ = −b/a and the product r₁ · r₂ = c/a. These are useful for checking answers without computing the roots directly.
How are quadratic equations used in real life?
Quadratics model projectile motion (throwing a ball), calculate areas for optimization problems, describe the shape of satellite dishes and parabolic mirrors, and appear in physics, engineering, economics (profit maximization), and computer graphics.