EqLab

Equation Solver

ax² + bx + c = 0, solved.

a (x²)

b (x)

Roots

3, 2

Discriminant: 1

Solves quadratics. For higher-order polynomials, use numerical methods.

What you entered

a: 1b: -5c: 6

1
Discriminant
-5² − 4×1×6= 1
2
Two real roots
x = (−-5 ± √1) ÷ (2×1)= 3, 2

Result

Roots: 3, 2

For 1x² + -5x + 6 = 0, the discriminant is 1, giving roots 3, 2.

Determinant of the coefficient matrix and what it means for solutions

For a 2x2 system, the determinant of the coefficient matrix decides whether a unique solution exists.

Coefficient matrix	Determinant	What it means
[[1, 2], [3, 4]]	-2	Unique solution
[[2, 0], [0, 2]]	4	Unique solution
[[4, 7], [2, 6]]	10	Unique solution
[[3, 8], [4, 6]]	-14	Unique solution
[[1, 2], [2, 4]]	0	No unique solution

The last row is the case to recognise: its second equation is just twice the first, so the two lines are parallel or identical and there is either no solution or infinitely many, never exactly one. A non-zero determinant guarantees the two lines cross at a single point. This is Cramer's rule territory - once the determinant is non-zero you can solve directly by substituting the constant column into each variable's position in turn.

Every quadratic has the same solving pattern

No matter the specific coefficients, ax² + bx + c = 0 is always solved by the same quadratic formula — that universality is exactly why it's one of the most memorized formulas in mathematics.

Reading the answer

Two real roots mean the equation crosses zero twice, a repeated root means it just touches zero at one point, and complex roots mean the curve never touches zero at all.

Frequently asked questions

Solve 2x² − 7x + 3 = 0 step by step

a = 2, b = −7, c = 3. Discriminant = (−7)² − 4(2)(3) = 49 − 24 = 25. Since 25 > 0, two real roots: x = (7 ± √25) / (2·2) = (7 ± 5) / 4. So x = 3 or x = 0.5.

How is this different from the polynomial roots calculator?

They solve the same equation but present results differently. The equation solver emphasizes the step-by-step quadratic formula process and factored form. The polynomial roots calculator emphasizes the geometric interpretation — where the curve crosses the x-axis. Same math, different emphasis.

When does a quadratic equation have no real solution?

When the discriminant (b² − 4ac) is negative. For example, x² + 1 = 0 has discriminant 0 − 4 = −4. The roots are complex numbers (x = ±i). The parabola never crosses the x-axis, sitting entirely above or below it.