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How to Solve Quadratic Equations
Three methods, clear worked examples, and a decision guide for picking the right approach every time.
By Chris Terry, Editor
Published June 7, 2026
Published June 7, 2026
Three methods, clear worked examples, and a decision guide for picking the right approach every time.
To solve a quadratic equation, plug the coefficients a, b, and c into the quadratic formula: x = (-b +/- sqrt(b^2 - 4ac)) / (2a). That formula always works. For simpler equations, factoring or completing the square can be faster.
A quadratic equation is any equation that can be written in the standard form:
where a, b, and c are real numbers and a is not zero. The highest power of x is 2, which is what makes it quadratic. According to Wolfram MathWorld, every quadratic equation has exactly two solutions (counting complex numbers), though those solutions may be equal or imaginary.
Real examples show up constantly: projectile motion, profit optimization, area problems. The shape they describe, the parabola, is everywhere in physics and engineering.
The quadratic formula solves any equation of the form ax^2 + bx + c = 0. No conditions, no special cases.
The +/- sign means you get two answers: one using addition, one using subtraction. The expression under the square root, b^2 - 4ac, is called the discriminant. It tells you what kind of solutions to expect before you finish the calculation.
You can check your work quickly with our quadratic formula calculator, which shows each step of the arithmetic.
Solve 2x^2 - 4x - 6 = 0.
First, read off the coefficients: a = 2, b = -4, c = -6.
Check by substituting back: 2(9) - 4(3) - 6 = 18 - 12 - 6 = 0. Correct.
Factoring is the fastest method when it works. The goal is to rewrite ax^2 + bx + c as a product of two binomials: (px + q)(rx + s) = 0. From there, each factor can equal zero separately, giving you the two roots.
Factoring works cleanly when the roots are integers or simple fractions. If the discriminant is not a perfect square, factoring over the integers is not possible and you should fall back to the formula.
Solve x^2 + 5x + 6 = 0.
Here a = 1, b = 5, c = 6. Look for two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3.
A quick check: (-2)^2 + 5(-2) + 6 = 4 - 10 + 6 = 0. Works.
Completing the square turns ax^2 + bx + c = 0 into a perfect square trinomial on one side. Move c to the right, divide through by a, then add (b / 2a)^2 to both sides. The left side becomes (x + b/2a)^2, which you can solve with a square root.
It is worth knowing because it is how the quadratic formula is derived. In practice, most people use the formula directly rather than completing the square from scratch each time. The method is also the basis for understanding how to find a derivative of polynomial functions, since manipulating standard forms is a shared skill.
Here is a simple decision guide.
The discriminant is your early-warning system. Calculate b^2 - 4ac before anything else. Positive means two real roots. Zero means one repeated root. Negative means no real roots.
Plug in a, b, and c to get both roots with full working steps.
The quadratic formula is x = (-b +/- sqrt(b^2 - 4ac)) / (2a). It gives the solutions to any equation in the form ax^2 + bx + c = 0, where a is not zero. The +/- means you get two solutions: one with addition and one with subtraction.
Calculate the discriminant: b^2 - 4ac. If the result is negative, the square root step produces an imaginary number. The equation has no real solutions in that case. It has two complex solutions instead.
The discriminant (b^2 - 4ac) tells you how many real solutions the equation has. A positive value means two distinct real roots. A value of zero means exactly one repeated root, sometimes called a double root. A negative value means no real roots at all, only complex ones.
Yes. Plot y = ax^2 + bx + c and find where the parabola crosses the x-axis. Each crossing point is a real solution. If the parabola sits entirely above or below the x-axis without touching it, the equation has no real solutions. Graphing is useful for a quick visual check, though it rarely gives exact values for irrational roots.
Editor at Encore Editorial, Chris Terry sets the editorial standards here and turns dense topics into plain English. He has written widely on education, finance, and consumer markets.