Quadratic equation
Solve a quadratic arranged as a*x^2 + b*x + c = 0. Move all terms to the left before entering the coefficients.
MAKE THE FORMULA CLEAR
Two linear equations
Solve equations arranged as a*x + b*y = c and d*x + e*y = f. Enter the six numbers, including minus signs, rather than pasting whole equations.
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Solution for x
2
Finds x and y, or reports no solution or infinitely many. Near-degenerate systems require a separate check.
Solution for y1
Determinant D = ae − bd. When nonzero, x = (ce − bf) ÷ D and y = (af − cd) ÷ D. Tests approximate floating-point zero after normalizing each row's scale. Near-degenerate systems may be unstable and are refused. For degenerate systems, checks rows for contradictions; 0x + 0y = 1 is not a valid identity.
Method and scope
Determinant D equals a*e minus b*d. When it is nonzero, x equals (c*e minus b*f) divided by D and y equals (a*f minus c*d) divided by D.
Degenerate equations can have no solution or infinitely many solutions. An impossible row such as 0x + 0y = 1 is not an identity.
Rows are normalized by scale for floating-point zero checks. A nearly degenerate system can be numerically unstable and may be rejected rather than showing an unreliable solution.
References:OpenStax: algebra and applications
What this helps you do
Solve equations arranged as a*x + b*y = c and d*x + e*y = f. Enter the six numbers, including minus signs, rather than pasting whole equations.
What to enter
Start with equation 1: x coefficient a, then fill the other relevant fields in the units shown. The field hints explain the required format and examples.
How the calculation works
Determinant D equals a*e minus b*d. When it is nonzero, x equals (c*e minus b*f) divided by D and y equals (a*f minus c*d) divided by D.
Worked example
For 2x + y = 5 and x - y = 1, enter 2,1,5 and 1,-1,1. The solution is x = 2 and y = 1.
Understanding the answer
Degenerate equations can have no solution or infinitely many solutions. An impossible row such as 0x + 0y = 1 is not an identity.
Scope and limitations
Rows are normalized by scale for floating-point zero checks. A nearly degenerate system can be numerically unstable and may be rejected rather than showing an unreliable solution.