Algebra Calculator - Quadratic & 2-Variable Linear System Solver
Free online Algebra Calculator. Solve quadratic equations ax² + bx + c = 0 and systems of 2 linear equations using real, repeated, or complex root logic.
AI Quick Summary
Definition & Purpose:
The Algebra Calculator solves quadratic equations ax^2 + bx + c = 0 (returning real, repeated, or complex roots) and 2-variable linear systems of equations a1x + b1y = c1 and a2x + b2y = c2 (using Cramer's Rule).
When to Use:
Use this tool to solve quadratic equations and 2-variable linear systems with step-by-step discriminant and determinant breakdowns.
Key Takeaway Insights:
- Quadratic Module computes discriminant D = b^2 - 4ac and solves real, repeated, or complex imaginary roots depending on its sign.
- Linear System Module uses Cramer's Rule to solve 2-variable linear systems (a1x + b1y = c1, a2x + b2y = c2).
- Identifies special system conditions: single unique solution, infinitely many solutions (coincident lines), or no solution (parallel lines).
Introduction
Algebra Calculator – Quadratic & Linear System Guide
Solving algebraic equations requires different mathematical methods depending on whether you are analyzing a second-degree polynomial (quadratic) or a 2-variable system of linear equations. This calculator supports two dedicated algebra modules, providing complete mathematical breakdowns for discriminants, determinants, and roots.
How the Algebra Modules Work
Module 1: Quadratic equations (ax^2 + bx + c = 0). The discriminant is D = b^2 - 4ac, and its sign determines the root type:
D > 0: quad x_1,2 = frac-b ± √(D)2a qquad D = 0: quad x = (-b / 2a) qquad D < 0: quad x_1,2 = (-b / 2a) ± ifrac√(-D)2a
Module 2: System of 2 linear equations. Solves a_1x + b_1y = c_1 and a_2x + b_2y = c_2 using Cramer's Rule:
D_s = a_1 b_2 - a_2 b_1 qquad D_x = c_1 b_2 - c_2 b_1 qquad D_y = a_1 c_2 - a_2 c_1
If D_s ≠ 0: unique solution x = D_x / D_s, y = D_y / D_s. If D_s = 0 and both D_x = 0 and D_y = 0: infinitely many solutions (coincident lines). If D_s = 0 and either D_x ≠ 0 or D_y ≠ 0: no solution (parallel lines).
Worked Examples
Example 1: Quadratic Mode — Two Real Roots (x^2 - 5x + 6 = 0)
With a = 1, b = -5, c = 6: discriminant D = (-5)^2 - 4(1)(6) = 25 - 24 = 1. Since D > 0: x_1 = (5 + √(1))/2 = 3, and x_2 = (5 - √(1))/2 = 2.
Example 2: Linear System Mode (2x + y = 5 and 3x - y = 5)
With a_1=2, b_1=1, c_1=5 and a_2=3, b_2=-1, c_2=5: D_s = 2(-1) - 3(1) = -5, D_x = 5(-1) - 5(1) = -10, D_y = 2(5) - 3(5) = -5. So x = -10/-5 = 2 and y = -5/-5 = 1.
Example 3: Quadratic Mode — Complex Roots (x^2 + 2x + 5 = 0)
With a = 1, b = 2, c = 5: discriminant D = 2^2 - 4(1)(5) = 4 - 20 = -16. Since D < 0, the roots are complex: real part = -2/(2 × 1) = -1, imaginary part = √(16)/(2 × 1) = 2. The roots are x_1,2 = -1 ± 2i.
What This Calculator Does Not Include
Frequently Asked Questions
What does the quadratic discriminant D indicate?
The discriminant D = b² − 4ac determines root behavior: if D > 0, there are 2 distinct real roots; if D = 0, there is 1 real repeated root; if D < 0, there are 2 complex imaginary roots (a ± bi).
What does a linear system determinant Ds = 0 mean?
When Ds = a1b2 − a2b1 = 0, the lines are parallel. If Dx = 0 and Dy = 0, the lines coincide (infinitely many solutions). If Dx ≠ 0 or Dy ≠ 0, the lines never intersect (no solution).
How does the calculator compute complex roots?
When the discriminant D is negative, the calculator cannot take a real square root of it, so it computes the real part as −b / (2a) and the imaginary part as √(−D) / (2a) (taking the square root of the positive value −D instead), then reports the two conjugate roots as Real Part plus or minus Imaginary Part times i, as shown in the x² + 2x + 5 = 0 example.
Why does the same Cramer's Rule formula work for any 2-variable linear system?
Cramer's Rule expresses the solution to a 2-by-2 linear system purely in terms of determinants built from the equations' coefficients, so it works for any pair of 2-variable linear equations without needing substitution or elimination steps — the calculator only needs the six coefficients (a1, b1, c1, a2, b2, c2) to compute Ds, Dx, and Dy directly.
Formula & Variables Explained
This tool utilizes standard equations formulated under standard rules.
Variables:
- Input parameter: Values supplied to resolve the output formula.
How to Calculate (Step-by-Step)
- Input the required parameters into the form.
- Click the calculate or auto-compute option.
- The outputs will refresh instantly with step-by-step variables.
Worked Examples Calculation
1Quadratic Equation with Two Real Roots (x^2 - 5x + 6 = 0)
Mode = Quadratic, a = 1, b = -5, c = 6
Discriminant D = (-5)^2 - 4(1)(6) = 25 - 24 = 1. Since D > 0, x1 = (5 + sqrt(1)) / 2 = 3; x2 = (5 - sqrt(1)) / 2 = 2.
Discriminant D = 1 | Root x1 = 3 | Root x2 = 2
2Linear System of 2 Equations (2x + y = 5, 3x - y = 5)
Mode = Linear System, a1 = 2, b1 = 1, c1 = 5; a2 = 3, b2 = -1, c2 = 5
Ds = 2(-1) - 3(1) = -5. Dx = 5(-1) - 5(1) = -10. Dy = 2(5) - 3(5) = -5. x = -10 / -5 = 2. y = -5 / -5 = 1.
System Determinant Ds = -5 | Variable x = 2 | Variable y = 1
3Quadratic Equation with Complex Roots (x^2 + 2x + 5 = 0)
Mode = Quadratic, a = 1, b = 2, c = 5
Discriminant D = 2^2 - 4(1)(5) = 4 - 20 = -16. Since D < 0, the roots are complex: Real Part = -2 / (2 x 1) = -1. Imaginary Part = sqrt(16) / (2 x 1) = 4 / 2 = 2.
Discriminant D = -16 | Root x1 = -1 + 2i | Root x2 = -1 - 2i
Real-World Applications
Widely used in student curriculum, professional projections, and quick estimations.
Limitations & Common Mistakes
- Entering incompatible unit formats (e.g. Mixing Metric and Imperial).
- Typographical mistakes in numeric entry fields.
Solves quadratic equations and 2-variable systems of linear equations only — it does not solve higher-degree polynomials or systems with 3 or more variables.
Frequently Asked Questions (FAQ)
Q:What does the quadratic discriminant D indicate?
The discriminant D = b^2 - 4ac determines root behavior: if D > 0, there are 2 distinct real roots; if D = 0, there is 1 real repeated root; if D < 0, there are 2 complex imaginary roots (a +/- bi).
Q:What does a linear system determinant Ds = 0 mean?
When Ds = a1b2 - a2b1 = 0, the lines are parallel. If Dx = 0 and Dy = 0, the lines coincide (infinitely many solutions). If Dx is not 0 or Dy is not 0, the lines never intersect (no solution).
Q:How does the calculator compute complex roots?
When the discriminant D is negative, the calculator cannot take a real square root of it, so it computes the real part as -b / (2a) and the imaginary part as sqrt(-D) / (2a) (taking the square root of the positive value -D instead), then reports the two conjugate roots as Real Part plus or minus Imaginary Part times i, as shown in the x^2 + 2x + 5 = 0 example.
Q:Why does the same Cramer's Rule formula work for any 2-variable linear system?
Cramer's Rule expresses the solution to a 2-by-2 linear system purely in terms of determinants built from the equations' coefficients, so it works for any pair of 2-variable linear equations without needing substitution or elimination steps — the calculator only needs the six coefficients (a1, b1, c1, a2, b2, c2) to compute Ds, Dx, and Dy directly.
References & Citations
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