Quadratic Formula

First bring the equation into the form ax² + bx + c = 0 and read off a, b and c. Then insert them into x = (−b ± √(b² − 4ac))/(2a). First compute the discriminant b² − 4ac under the root, take the root and form the two solutions with the plus and the minus sign. Example: 2x² − 4x − 6 = 0 gives x = (4 ± √(16 + 48))/4 = (4 ± 8)/4, that is x₁ = 3 and x₂ = −1. Watch the signs of b and c and do not forget the ±, otherwise you lose one solution.

The discriminant D = b² − 4ac sits under the root and decides the number and type of real solutions. If D is greater than zero, two distinct real solutions exist, because the root gives a positive value that is once added and once subtracted. If D equals zero, both solutions merge into a single double root x = −b/(2a). If D is less than zero, there is no real solution, only two complex ones, because you cannot take a real root of a negative number. So from the sign of the discriminant alone you can judge the solvability before calculating further.

Both formulas solve quadratic equations but assume different starting forms. The quadratic or ABC formula works with the general form ax² + bx + c = 0 and gives x = (−b ± √(b² − 4ac))/(2a); the leading factor a may be arbitrary. The pq formula, by contrast, requires the normalized form x² + px + q = 0 with leading coefficient one and gives x = −p/2 ± √((p/2)² − q). If you want to use the pq formula, you first divide the equation by a. Both lead to the same result; the choice is a matter of taste and often depends on which form the problem provides.

If the discriminant b² − 4ac equals zero, the root vanishes and the two solutions of the formula merge into a single one: x = −b/(2a). This is called a double root. Geometrically it means that the associated parabola does not cross the x-axis but only touches it; the vertex lies exactly on the axis. This special situation occurs in problems about tangency of curves or in limiting cases, for example when a line just touches a parabola. If you recognize D = 0, you no longer need the ±, because both signs give the same result.

Vieta formulas link the two solutions x₁ and x₂ of a quadratic equation x² + px + q = 0 directly with the coefficients: the sum of the solutions is x₁ + x₂ = −p, their product is x₁·x₂ = q. With this you can often find solutions in your head or check a result without going through the whole formula. If you find two numbers whose sum is −p and whose product is q, those are exactly the roots. Example: x² − 5x + 6 = 0 has solutions with sum 5 and product 6, that is 2 and 3. Vieta is excellent for quickly checking the values you computed with the formula.