Equation Solver
Solve linear, quadratic, and systems of equations step by step. Get solutions with detailed explanations for algebra problems including the quadratic formula.
Select Equation Type:
Linear Equation: ax + b = c
Equation Solving Tips
- • Linear equations have exactly one solution when a ≠ 0
- • Quadratic equations can have 0, 1, or 2 real solutions based on the discriminant
- • Systems of equations intersect at a point, are parallel, or coincident
- • The quadratic formula works for all quadratic equations
- • Cramer's rule uses determinants to solve systems efficiently
Related Calculators
About This Calculator
Solving equations is one of the fundamental skills in algebra and mathematics. Whether you're working with simple linear equations, quadratic equations, or systems of multiple equations, this calculator provides step-by-step solutions to help you understand the solving process and verify your work.
Types of Equations You Can Solve:
- Linear Equations (ax + b = c): Single-variable equations with degree 1
- Quadratic Equations (ax² + bx + c = 0): Polynomial equations with degree 2
- Systems of Linear Equations: Two equations with two unknowns
Why Step-by-Step Solutions Matter: Simply getting an answer isn't enough for learning. This calculator shows every step of the solving process, helping you understand the underlying methods and principles. Whether you're checking homework, preparing for exams, or learning new concepts, seeing the work is essential.
Mathematical Methods Used:
- Algebraic manipulation for linear equations
- The quadratic formula for second-degree polynomials
- Cramer's rule (determinants) for systems of equations
This calculator is perfect for algebra students, test preparation, and anyone who needs to solve equations quickly while understanding the process. For graphing these equations, try our Graphing Calculator. For more advanced polynomial work, see our Polynomial Calculator.
How to Use the Equation Solver
- 1Select the type of equation you want to solve (linear, quadratic, or system).
- 2For linear equations, enter the coefficients a, b, and c for ax + b = c.
- 3For quadratic equations, enter a, b, and c for ax² + bx + c = 0.
- 4For systems, enter all six coefficients for the two equations.
- 5Review the solution(s) displayed in the result area.
- 6Study each step to understand how the solution was derived.
- 7For quadratic equations, check the discriminant analysis.
- 8For systems, verify the solution satisfies both equations.
- 9Use the solution to check your own work or learn the method.
- 10Try different values to see how changing coefficients affects solutions.
Solving Linear Equations
Linear equations are the foundation of algebra.
Standard Form
ax + b = c where:
- a = coefficient of x (cannot be 0)
- b = constant added to the variable term
- c = the value the expression equals
Solving Method
Goal: Isolate x on one side of the equation
Steps:
- Subtract b from both sides: ax = c - b
- Divide both sides by a: x = (c - b) / a
Example
Solve: 3x + 7 = 19
- Original: 3x + 7 = 19
- Subtract 7: 3x = 19 - 7 = 12
- Divide by 3: x = 12 / 3 = 4
Verification: 3(4) + 7 = 12 + 7 = 19 ✓
Special Cases
When a = 0:
- 0x + b = c becomes b = c
- If b = c: Infinitely many solutions
- If b ≠ c: No solution
When b = 0:
- ax = c simplifies to x = c/a
Common Mistakes to Avoid
- Forgetting to perform the same operation on both sides
- Sign errors when subtracting negative numbers
- Not simplifying fractions in the final answer
Solving Quadratic Equations
Quadratic equations have the variable raised to the second power.
Standard Form
ax² + bx + c = 0 where:
- a ≠ 0 (otherwise it's linear)
- b = coefficient of x (can be 0)
- c = constant term (can be 0)
The Quadratic Formula
For any quadratic equation ax² + bx + c = 0:
x = (-b ± √(b² - 4ac)) / 2a
This formula always works, regardless of whether the equation factors nicely.
The Discriminant (Δ = b² - 4ac)
The discriminant determines the nature of solutions:
| Discriminant | Solutions |
|---|---|
| Δ > 0 | Two distinct real roots |
| Δ = 0 | One repeated real root |
| Δ < 0 | Two complex conjugate roots |
Example: Two Real Roots
Solve: x² - 5x + 6 = 0
- a = 1, b = -5, c = 6
- Δ = (-5)² - 4(1)(6) = 25 - 24 = 1
- x = (5 ± √1) / 2 = (5 ± 1) / 2
- x₁ = 6/2 = 3, x₂ = 4/2 = 2
Verification: (3)² - 5(3) + 6 = 9 - 15 + 6 = 0 ✓
Alternative Methods
Factoring: When ax² + bx + c = a(x - r₁)(x - r₂) Completing the square: Useful for deriving the formula Graphing: Solutions are x-intercepts of the parabola
Systems of Linear Equations
A system of equations contains multiple equations with multiple unknowns.
2×2 System
a₁x + b₁y = c₁ a₂x + b₂y = c₂
Solution Methods
1. Substitution: Solve one equation for one variable, substitute into the other
2. Elimination: Add/subtract equations to eliminate one variable
3. Cramer's Rule: Use determinants (most efficient for 2×2)
Cramer's Rule
Calculate determinants:
- D = a₁b₂ - a₂b₁ (main determinant)
- Dₓ = c₁b₂ - c₂b₁
- Dᵧ = a₁c₂ - a₂c₁
Solutions: x = Dₓ/D, y = Dᵧ/D
Example
Solve: 2x + 3y = 8 x - y = 1
- D = (2)(-1) - (1)(3) = -2 - 3 = -5
- Dₓ = (8)(-1) - (1)(3) = -8 - 3 = -11
- Dᵧ = (2)(1) - (1)(8) = 2 - 8 = -6
x = -11/-5 = 2.2, y = -6/-5 = 1.2
Types of Solutions
| Condition | Geometric Meaning | Solutions |
|---|---|---|
| D ≠ 0 | Lines intersect | One unique solution |
| D = 0, consistent | Lines coincide | Infinitely many |
| D = 0, inconsistent | Lines are parallel | No solution |
The Discriminant in Depth
The discriminant is a powerful tool for analyzing quadratic equations without solving them.
Definition
For ax² + bx + c = 0: Δ = b² - 4ac
Geometric Interpretation
The discriminant relates to the parabola y = ax² + bx + c:
Δ > 0: Parabola crosses x-axis at two points Δ = 0: Parabola touches x-axis at one point (vertex) Δ < 0: Parabola doesn't cross x-axis
Using the Discriminant
To quickly determine root nature:
| Equation | Δ | Nature |
|---|---|---|
| x² - 4x + 3 = 0 | 16 - 12 = 4 | Two real roots |
| x² - 4x + 4 = 0 | 16 - 16 = 0 | One repeated root |
| x² - 4x + 5 = 0 | 16 - 20 = -4 | Complex roots |
Perfect Square Discriminant
When Δ is a perfect square (1, 4, 9, 16, ...), the quadratic factors nicely over integers.
Example: x² - 5x + 6 = 0
- Δ = 25 - 24 = 1 (perfect square)
- Factors as (x - 2)(x - 3) = 0
Complex Roots
When Δ < 0, roots are complex conjugates: x = (-b ± i√|Δ|) / 2a
The real part is -b/2a, imaginary part is √|Δ|/2a.
Common Equation Forms
Recognizing equation forms helps choose the best solving method.
Linear Forms
Standard form: ax + b = c Slope-intercept: y = mx + b Point-slope: y - y₁ = m(x - x₁) Two-point: (y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)
Quadratic Forms
Standard: ax² + bx + c = 0 Vertex: a(x - h)² + k = 0 (vertex at (h, k)) Factored: a(x - r₁)(x - r₂) = 0 (roots r₁, r₂)
Converting Between Forms
Standard to factored:
- Find roots using quadratic formula
- Write as a(x - r₁)(x - r₂)
Standard to vertex:
- Complete the square
- h = -b/2a, k = c - b²/4a
Example: x² - 6x + 8 = 0
- Factored: (x - 2)(x - 4) = 0
- Vertex: (x - 3)² - 1 = 0
Absolute Value Equations
|ax + b| = c has two cases:
- ax + b = c → x = (c - b)/a
- ax + b = -c → x = (-c - b)/a
Radical Equations
√(ax + b) = c
- Square both sides: ax + b = c²
- Solve: x = (c² - b)/a
- Always verify (squaring can introduce extraneous solutions)
Applications of Equation Solving
Equations model countless real-world situations.
Physics Applications
Kinematics:
- Distance: d = vt (linear)
- Projectile height: h = -16t² + v₀t + h₀ (quadratic)
Example: Ball thrown upward at 64 ft/s from ground h = -16t² + 64t = 0 -16t(t - 4) = 0 Lands at t = 4 seconds
Finance Applications
Break-even analysis: Revenue = Cost 50x = 20x + 1500 30x = 1500 x = 50 units to break even
Mixture problems: 0.05(x) + 0.20(100 - x) = 0.10(100) 0.05x + 20 - 0.20x = 10 -0.15x = -10 x = 66.7 mL of 5% solution
Geometry Applications
Area problems: Rectangle with perimeter 20, area 24 2l + 2w = 20, lw = 24 l + w = 10, l = 10 - w (10 - w)w = 24 w² - 10w + 24 = 0 w = 4 or 6 (dimensions are 4×6)
Rate Problems
Work problems: 1/A + 1/B = 1/T (combined rate)
Distance problems: d₁ = r₁t, d₂ = r₂t When do they meet? Set d₁ + d₂ = total distance
Pro Tips
- 💡Always verify your solution by substituting back into the original equation.
- 💡For quadratics, calculate the discriminant first to know what type of roots to expect.
- 💡When factoring, look for common factors before trying other methods.
- 💡In systems of equations, choose the method that minimizes arithmetic.
- 💡Complex roots always come in conjugate pairs (a + bi and a - bi).
- 💡The sum of quadratic roots is -b/a; the product is c/a.
- 💡Graph equations to visualize solutions and check reasonableness.
- 💡When the discriminant is a perfect square, the quadratic factors nicely.
- 💡For word problems, define variables clearly before writing equations.
- 💡Remember that dividing by a variable might introduce restrictions (variable ≠ 0).
- 💡Systems with D = 0 are either parallel lines (no solution) or the same line (infinite solutions).
- 💡Practice mental math with common patterns like (x-a)(x+a) = x² - a².
Frequently Asked Questions
The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It solves any quadratic equation ax² + bx + c = 0. The ± symbol means you calculate two values: one using + and one using -, giving you both roots (solutions). The expression b² - 4ac under the square root is called the discriminant and determines whether roots are real or complex.
