Math & Statistics
Algebra Equation Solver – Solve Linear & Quadratic Equations Online
Use our free Algebra Equation Solver to quickly solve linear, quadratic, and polynomial equations. Get step-by-step solutions and check your answers instantly for better understanding.
How to Solve Algebraic Equations
An algebraic equation is a mathematical statement where two expressions are equal, containing one or more unknown variables. The goal is to find the value(s) of the variable that make the equation true.
Linear equations (ax + b = 0) have one solution: x = −b/a. Quadratic equations (ax² + bx + c = 0) can have 0, 1, or 2 real solutions, found using the quadratic formula: x = (−b ± √(b²−4ac)) / 2a. The discriminant (b²−4ac) determines the number of solutions.
Linear Equation: ax + b = 0
Example: For 2x − 6 = 0, enter a = 2, b = −6
Quadratic Equation: ax² + bx + c = 0
Example: For x² − 5x + 6 = 0, enter a = 1, b = −5, c = 6
How to Use This Calculator
- Select the equation type (Linear or Quadratic)
- Enter the coefficients for your equation
- Click Solve Equation to see the solution
- Review the step-by-step breakdown and verification
Common Use Cases
- Algebra homework
- Physics equations
- Engineering problems
- Financial modeling
- Exam preparation
- Projectile motion
Common Equation Types & Formulas
| Type | Standard Form | Solution Formula | Example |
|---|---|---|---|
| Linear | ax + b = 0 | x = −b / a | 3x + 9 = 0 → x = −3 |
| Quadratic | ax² + bx + c = 0 | x = (−b ± √(b²−4ac)) / 2a | x² − 5x + 6 = 0 → x = 2, 3 |
| Two-step | ax + b = c | x = (c − b) / a | 2x + 3 = 11 → x = 4 |
| Proportion | a/b = c/x | x = bc / a | 3/4 = 6/x → x = 8 |
Understanding the Discriminant (Δ = b² − 4ac)
For quadratic equations, the discriminant determines the nature and number of solutions:
| Discriminant Value | Number of Solutions | Nature | Graph Behavior |
|---|---|---|---|
| Δ > 0 | 2 distinct real solutions | Two different x-intercepts | Parabola crosses x-axis twice |
| Δ = 0 | 1 repeated real solution | One x-intercept (vertex touches axis) | Parabola touches x-axis once |
| Δ < 0 | No real solutions | Two complex conjugate solutions | Parabola doesn't cross x-axis |
Step-by-Step Examples
Linear: 4x + 12 = 0
- Subtract 12: 4x = −12
- Divide by 4: x = −12 ÷ 4
- Solution: x = −3
Quadratic: x² − 7x + 10 = 0
- a=1, b=−7, c=10
- Δ = 49 − 40 = 9
- x = (7 ± 3) / 2
- Solutions: x = 5, x = 2
FAQ – Algebra Equation Solver
What is a linear equation and how do I solve it?
A linear equation has the form ax + b = 0, where x appears only to the first power. To solve: isolate x by subtracting b from both sides, then divide by a. The solution is x = −b/a. Example: 3x + 9 = 0 → 3x = −9 → x = −3. Linear equations always have exactly one solution (unless a = 0).
What is the quadratic formula?
The quadratic formula solves any equation of the form ax² + bx + c = 0: x = (−b ± √(b²−4ac)) / 2a. The ± means you calculate twice: once with + and once with −, giving up to two solutions. This formula works even when factoring is difficult or impossible.
What does the discriminant tell me?
The discriminant (Δ = b² − 4ac) reveals how many real solutions exist. If Δ > 0: two distinct real solutions. If Δ = 0: one repeated solution (the parabola touches the x-axis). If Δ < 0: no real solutions (only complex numbers). Calculate the discriminant first to know what to expect.
What if coefficient 'a' equals zero?
If a = 0 in a quadratic equation, it becomes linear (bx + c = 0). If a = 0 in a linear equation (0x + b = 0), then: if b = 0, there are infinite solutions (0 = 0 is always true); if b ≠ 0, there's no solution (e.g., 0x + 5 = 0 means 5 = 0, which is false).
Can quadratic equations have no solution?
Quadratic equations always have solutions, but they may not be real numbers. When the discriminant is negative, the solutions are complex numbers involving i (the imaginary unit, where i² = −1). For example, x² + 1 = 0 has solutions x = ±i. This calculator shows complex solutions in such cases.
How do I convert an equation to standard form?
Move all terms to one side so the equation equals zero. Example: 2x² + 3 = 5x → 2x² − 5x + 3 = 0. Now you can identify a = 2, b = −5, c = 3. Always arrange terms in descending order of powers (x², then x, then constant).
What's the difference between solving and factoring?
Factoring rewrites the equation as a product: x² − 5x + 6 = (x − 2)(x − 3) = 0. Then set each factor to zero: x = 2 or x = 3. The quadratic formula always works, but factoring is faster when possible. Not all quadratics factor nicely with integers.
How do I verify my solution is correct?
Substitute your answer back into the original equation. If both sides are equal, the solution is correct. Example: For 2x + 6 = 0, if x = −3, check: 2(−3) + 6 = −6 + 6 = 0 ✓. For quadratics, verify both solutions separately.
What are complex solutions?
Complex solutions occur when the discriminant is negative. They have the form a + bi, where i = √(−1). Example: x² + 4 = 0 → x = ±2i. Complex solutions always come in conjugate pairs (a + bi and a − bi). They're used in engineering, physics, and advanced mathematics.
Can this solver handle equations with fractions?
Yes, enter decimal equivalents for fractions. For example, if your equation is (1/2)x² − (3/4)x + (1/8) = 0, enter a = 0.5, b = −0.75, c = 0.125. Alternatively, multiply the entire equation by the LCD to eliminate fractions first, then enter the integer coefficients.