QuadSolve

Quadratic Formula Calculator

x = (−b ± √(b²−4ac)) / (2a).

Discriminant

x = (−b ± √(b²−4ac)) / (2a)

Roots: 4, -1

What you entered

a: 1b: -3c: -4

1
Discriminant
-3² − 4×1×-4= 25
2
Two real roots
x = (−-3 ± √25) ÷ (2×1)= 4, -1

Result

Discriminant: 25

For 1x² + -3x + -4 = 0, the discriminant is 25, giving roots 4, -1.

Quadratic roots and what the discriminant tells you

For ax squared plus bx plus c, the discriminant b squared minus 4ac decides how many real roots exist before you solve anything.

Equation	Discriminant	Roots	Type
x2 - 3x + 2	1	2, 1	Two real roots
x2 - 5x + 6	1	3, 2	Two real roots
x2 - 4	16	2, -2	Two real roots
2x2 + 5x - 3	49	0.5, -3	Two real roots
x2 + 2x + 1	0	-1	One repeated root
x2 + x + 1	-3	-0.5 +- 0.866i	Two complex roots

The discriminant column is the shortcut: positive means the parabola crosses the x-axis twice, zero means it touches at exactly one point, and negative means it never touches and the roots are complex. You can read that off before doing any arithmetic. A discriminant that is a perfect square, as in the first four rows, means the roots are rational and the equation would also factor neatly. The complex roots in the last row always come in conjugate pairs when the coefficients are real, which is why they are written with a plus-or-minus.

The discriminant decides the outcome

The value under the square root (b² − 4ac) tells you what kind of roots to expect: positive means two real roots, zero means one repeated root, and negative means two complex roots.

Real-world quadratics

Anything involving projectile motion, the shape of a satellite dish, or maximizing area/profit under a constraint tends to produce a quadratic equation — this formula solves all of them exactly.

Frequently asked questions

How do I use the quadratic formula?

For ax² + bx + c = 0: x = (−b ± √(b²−4ac)) / (2a). Identify a, b, c from your equation, compute the discriminant (b²−4ac), then plug in. For 2x² + 5x − 3 = 0: a=2, b=5, c=−3, discriminant=49, x = (−5±7)/4, so x = 0.5 or x = −3.

What does the discriminant tell me?

b²−4ac > 0: two distinct real roots. b²−4ac = 0: one repeated root (the parabola touches the x-axis). b²−4ac < 0: two complex roots (the parabola doesn't cross the x-axis). The discriminant determines the nature of solutions without solving.

When should I use the quadratic formula vs factoring?

Try factoring first — it's faster when it works (integer roots). Use the quadratic formula when the equation doesn't factor neatly, has irrational or complex roots, or when you want guaranteed results. The formula always works for any quadratic equation.