Quadratic Equation: How to Solve It and Graph the Parabola

A quadratic equation stands out by a term with x squared: in standard form it’s written ax² + bx + c = 0, with a, b, and c real numbers and a not equal to zero. If a were zero, the quadratic term would disappear and the equation would drop to a linear one. Anyone who has already met linear equations will recognize the same setup here — solving means finding the values of x that make the equality true — but the method differs because the degree is higher.

Three methods for finding the solutions

Factoring breaks the trinomial ax² + bx + c into a product of two first-degree factors and applies the zero-product property: if a product of two factors is zero, at least one of them must be zero. Setting each factor equal to zero gives the solutions of the equation. This works when the trinomial factors easily; equations that don’t factor at a glance need one of the other two methods.

Completing the square adds or subtracts terms on both sides of the equation until one side becomes a perfect square trinomial, a form that can be written as the square of a binomial. From there, the square root property applies and x can be isolated. The method works directly when the coefficient of the quadratic term is 1; otherwise the whole equation is divided by that coefficient first. Anyone already working with symbolic manipulation in algebra will recognize the same logic here, applied to an equation instead of an expression.

The quadratic formula works for every quadratic equation, no exceptions:

Method	When to use it	How it works
Factoring	The trinomial factors easily	Zero-product property: each factor set to zero gives one solution
Completing the square	Coefficient of x² equal to 1 (or divided down to 1)	One side becomes a perfect square, then x is isolated
Quadratic formula	Always, for any equation	x = (-b ± √(b² - 4ac)) / 2a

Practical example: solve 2x² - 4x - 6 = 0. Dividing everything by 2 gives x² - 2x - 3 = 0, with a = 1, b = -2, c = -3. The discriminant is Δ = b² - 4ac = 4 - 4·1·(-3) = 4 + 12 = 16. The square root of 16 is 4, so x = (2 ± 4) / 2: the first solution is x = 3, the second is x = -1.

The discriminant: what the sign tells you

The discriminant Δ = b² - 4ac sits inside the square root of the quadratic formula, and its sign tells you in advance how many solutions to expect and of what kind, before the calculation is even finished.

Sign of the discriminant	Solutions of the equation
Positive	Two distinct real solutions (rational if Δ is a perfect square, irrational otherwise)
Zero	One real solution, counted twice
Negative	Two complex conjugate solutions

Anyone who has already met complex numbers will recognize the point where the equation genuinely needs them: when the discriminant is negative, no real number squared gives a negative number, so the square root of the discriminant is written with the imaginary unit.

The parabola: vertex, axis, and concavity

The equation ax² + bx + c = 0 has a sibling equation, y = ax² + bx + c, which describes not just the points where y is zero but the entire curve on the coordinate plane: a parabola. Written in vertex form, f(x) = a(x - h)² + k, the pair (h, k) is the curve’s vertex, the point where the parabola changes direction. The vertex’s x-coordinate comes from h = -b/(2a), and its y-coordinate k is the value of the function at that point.

Both the concavity and the type of vertex depend on the sign of a. Going back to the earlier example, y = 2x² - 4x - 6 has a = 2, so h = -(-4)/(2·2) = 1 and k = f(1) = 2 - 4 - 6 = -8: the vertex is (1, -8), and because a is positive, that point is a minimum, with both branches of the curve climbing away from it. With a negative coefficient the opposite happens: both branches drop away from the vertex, which becomes the highest point, a maximum.

The axis of symmetry is the vertical line through the vertex that splits the parabola into two mirror halves: every point on the curve has a matching point at the same height on the other side of the axis. That’s why the two solutions of an equation with a positive discriminant always sit the same distance from the vertex’s x-coordinate, one to the left and one to the right.

Focus and directrix: the geometric definition

The parabola also comes from a purely geometric definition, independent of its coefficients. It’s one of the conic sections: it comes from cutting a circular cone with a plane parallel to one of its generating lines. The fuller study of this curve, alongside the other conics, belongs to analytic geometry.

Given a fixed point F, the focus, and a fixed line d, the directrix, every point on the parabola sits at exactly the same distance from F and from d. In this definition, the axis of symmetry connects the focus to the directrix while staying perpendicular to it; the vertex sits halfway along that path, where the axis touches the curve.

The discriminant of the trinomial ax² + bx + c plays the same role when the parabola is read as a graph: its sign determines where the curve meets the x-axis. A positive discriminant means the parabola crosses the x-axis at two distinct points; a discriminant of zero means it just touches the axis at one point, tangent to it; a negative discriminant means it never touches the axis at all. The two readings — the algebraic one about the solutions and the geometric one about the x-axis intersection — describe the same fact from two angles.