AlgebraEdge

Algebra Solver

Drop in the coefficients — see the roots, discriminant, and the factored form, with steps.

1x² + -3x + -4 = 0

x₁ = 4, x₂ = -1

1Discriminant Δ = b² − 4ac = -3² − 4·1·-4 = 25
2Apply x = (−b ± √Δ) / (2a) with a=1, b=-3, c=-4
3Two real roots (or one repeated if Δ = 0).

Roots and discriminants for common textbook quadratics

Every row is the quadratic mode of this solver: enter the three coefficients a, b and c for ax^2 + bx + c = 0 and it returns the discriminant b^2 - 4ac followed by the roots.

Equation	a, b, c to enter	Discriminant	Roots
x^2 - 3x - 4 = 0	1, -3, -4	25	x1 = 4, x2 = -1
x^2 - 5x + 6 = 0	1, -5, 6	1	x1 = 3, x2 = 2
x^2 - 4 = 0	1, 0, -4	16	x1 = 2, x2 = -2
x^2 - 2x - 8 = 0	1, -2, -8	36	x1 = 4, x2 = -2
2x^2 - 4x - 6 = 0	2, -4, -6	64	x1 = 3, x2 = -1
3x^2 + 5x - 2 = 0	3, 5, -2	49	x1 = 0.333333, x2 = -2
6x^2 + 11x - 35 = 0	6, 11, -35	961	x1 = 1.666667, x2 = -3.5
x^2 + 2x + 1 = 0	1, 2, 1	0	x = -1 (repeated)
x^2 + x + 1 = 0	1, 1, 1	-3	x = -0.5 +/- 0.866025i
x^2 + 4x + 5 = 0	1, 4, 5	-4	x = -2 +/- 1i

A positive discriminant gives two distinct real roots, zero gives one repeated root, and a negative value gives a conjugate pair of complex roots, which is the last two rows. Roots are displayed to six decimal places, so 1/3 appears as 0.333333. Switch to linear mode for ax + b = 0, where the single root is always x = -b/a.

Reading the discriminant

For a quadratic, the discriminant b²−4ac tells you what kind of roots to expect. Positive: two real roots. Zero: one repeated real root. Negative: two complex conjugate roots.

The quadratic formula

x = (−b ± √(b²−4ac)) / 2a. This works for any quadratic equation ax² + bx + c = 0, regardless of whether it factors neatly. The ± means there are usually two solutions — plug in + for one root and − for the other.

Frequently asked questions

How do I solve a quadratic equation?

Three methods: (1) Factoring — if ax² + bx + c factors into (px + q)(rx + s) = 0, set each factor to zero. (2) Quadratic formula — x = (−b ± √(b²−4ac)) / 2a, always works. (3) Completing the square — rewrite as (x + h)² = k, then x = −h ± √k. This calculator uses the quadratic formula and shows every step.

What does 'no real solution' mean?

When the discriminant (b²−4ac) is negative, the square root of a negative number doesn't exist in real numbers. The equation has two complex conjugate solutions: x = (−b ± i√|b²−4ac|) / 2a. Graphically, the parabola doesn't cross the x-axis.

How do I solve a linear equation?

For ax + b = 0: subtract b from both sides, then divide by a. x = −b/a. Example: 3x + 12 = 0 → 3x = −12 → x = −4. Linear equations always have exactly one solution (unless a = 0, in which case it's either no solution or infinitely many).