Equation Solver
Solve the equation step by step — linear and quadratic equations with every step explained.
Enter Your Equation
Enter your equation using standard mathematical notation
Solution
x₁
Real number
x₂
Real number
Step-by-Step Solution
Step 1: Original equation
Step 2: Expand both sides
Distribute, expand products, and combine like terms on each side
Step 3: Move everything to the left side
Subtract the right side from both sides
Step 4: Identify coefficients
Standard form: x² - 5x + 6 = 0
Step 5: Calculate the discriminant
Δ > 0 — two distinct real solutions
Step 6: Apply the quadratic formula
Step 7: Solutions
Step 8: Verify by substitution
Both solutions balance the original equation
Parabola Properties
Input Tips
- • Use ^ for exponents (e.g., x^2 for x squared) — ² and ³ also work
- • Equations must have exactly one equals sign
- • Implicit multiplication is recognized: 2x, 2(x+1), (x+1)(x+2)
- • Division by numbers works anywhere: x/2 + 3 = 5
- • Supported: linear (ax+b=c) and quadratic (ax²+bx+c=0) equations
- • Complex solutions are shown for negative discriminants
Quick examples
How to Solve the Equation 2x + 6 = 14, Step by Step
This is exactly what the solver above shows when you type in 2x + 6 = 14 — every step, plus a verification that the answer is right:
- Step 1 — Original equation:2x + 6 = 14
- Step 2 — Move everything to the left side:2x - 8 = 0Subtract the right side from both sides
- Step 3 — Solve for x:2x = 8 → x = 8 / 2Move the constant to the right side, then divide both sides by 2
- Step 4 — Solution:x = 4
- Step 5 — Verify by substitution:2(4) + 6 = 14 ✓Both sides of the original equation match — the solution checks out
The same process works for any linear equation, and quadratics get the full quadratic-formula treatment: coefficients, discriminant, both roots, and verification.
Understanding Equation Solving
Solving equations is a fundamental skill in algebra that involves finding the value(s) of unknown variables that make the equation true. Different types of equations require different solving strategies, from simple algebraic manipulation to advanced formulas.
Types of Equations
Linear Equations
Form: ax + b = c
Graph: Straight line
Solutions: Always one solution (unless degenerate)
Method: Algebraic manipulation
Quadratic Equations
Form: ax² + bx + c = 0
Graph: Parabola
Solutions: 0, 1, or 2 solutions
Method: Factoring, formula, or completing square
The Importance of Standard Form
Before solving any equation, it\'s crucial to convert it to standard form by moving all terms to one side. This makes it easier to identify the equation type and apply the appropriate solving method.
Linear Equation Solving Guide
Basic Steps
- Simplify both sides by combining like terms
- Move all variable terms to one side
- Move all constant terms to the other side
- Divide by the coefficient of the variable
- Check your solution by substituting back
Example: Solve 3x + 7 = 2x + 15
Quadratic Equation Methods
Factoring
Best when factors are integers
(x - 2)(x - 3) = 0
x = 2 or x = 3
Quadratic Formula
Works for any quadratic
Always gives exact answer
Handles complex roots
Completing Square
Useful for vertex form
(x + 3)² - 4 = 0
x = -3 ± 2
Understanding the Discriminant
For quadratic equations ax² + bx + c = 0, the discriminant Δ = b² - 4ac determines the nature of solutions:
Δ > 0
Two distinct real roots
Δ = 25 - 24 = 1
Roots: 3 and 2
Δ = 0
One repeated real root
Δ = 16 - 16 = 0
Root: 2 (double)
Δ < 0
Two complex conjugate roots
Δ = 4 - 20 = -16
Roots: -1 ± 2i
Problem-Solving Strategies
Before You Start
- ✓ Identify the type of equation
- ✓ Check if it's already in standard form
- ✓ Look for common factors to simplify
- ✓ Decide on the best solving method
- ✓ Estimate the answer if possible
Common Pitfalls
- ✗ Forgetting to change signs when moving terms
- ✗ Not distributing negative signs properly
- ✗ Dividing by zero or potential zero
- ✗ Losing solutions when simplifying
- ✗ Not checking answers in original equation
Applications in Real Life
Linear Equations
- • Cost analysis and budgeting
- • Distance, rate, and time problems
- • Temperature conversions
- • Simple interest calculations
- • Manufacturing and production
Quadratic Equations
- • Projectile motion in physics
- • Area and optimization problems
- • Economics and profit maximization
- • Engineering and design
- • Computer graphics and gaming
Study Tips for Equation Solving
- Practice regularly: Start with simple equations and gradually increase complexity. Consistency is key to mastering algebraic manipulation.
- Show all steps: Even when you can solve mentally, write out each step. This helps catch errors and builds good habits.
- Check your work: Always substitute your answer back into the original equation. This simple step catches most errors.
- Understand, don\'t memorize: Focus on understanding why each step works rather than memorizing procedures.
- Use multiple methods: Try solving the same equation different ways to deepen your understanding.
How it works
A math equation solver isolates the unknown by applying inverse operations to both sides until the variable stands alone. Linear equations take one rearrangement; quadratics use factoring or the quadratic formula; systems solve several equations at once.
Solve by inverse operations
Apply the same inverse operation to both sides until the variable is isolated
- variable
- the unknown being solved for
- inverse op
- undoes what's done to the variable
Worked example
- Solve 5x − 4 = 16
- Add 4: 5x = 20
- Divide by 5: x = 4
x = 4.
Good to know
- Keep the equation balanced — every operation applies to both sides.
- Quadratics (x²) need factoring or the quadratic formula, not just rearranging.
- Always verify by substituting the solution back in.
Related Calculators
Frequently Asked Questions
How do I solve an equation?
Isolate the variable by applying the same inverse operation to both sides until it stands alone. For 2x + 6 = 14: subtract 6 from both sides (2x = 8), then divide both sides by 2 (x = 4). Type any equation into the solver above and it shows every one of these steps for you.
How do I solve for x?
"Solve for x" means finding the value of x that makes both sides equal. Move every term containing x to one side and every constant to the other, combine like terms, then divide by the coefficient of x. If the equation has an x² term, use factoring or the quadratic formula instead. Always check the answer by substituting it back in — the solver does this verification automatically.
What is the quadratic formula and when is it used?
The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) solves any quadratic equation ax² + bx + c = 0. It's used when factoring is difficult or impossible. The discriminant (b² - 4ac) determines whether solutions are real or complex: positive means two real solutions, zero means one repeated solution, negative means two complex solutions.
What types of equations can this equation solver handle?
The solver handles linear equations (ax + b = c) and quadratic equations (ax² + bx + c = 0), including parentheses, fractions with numeric denominators, and implicit multiplication like 2(x - 3). It provides real and complex solutions with step-by-step explanations, and every solution is verified by substituting it back into the original equation.
How do I enter equations correctly?
Enter equations using standard mathematical notation: use ^ for exponents (x^2 for x squared), * for multiplication (though 2x is recognized), and standard +, -, = symbols. Equations must have exactly one equals sign. Examples: '2x + 5 = 13' or 'x^2 - 5x + 6 = 0'.
What are complex solutions?
Complex solutions occur when quadratic equations have no real roots (negative discriminant). They're expressed as a + bi, where 'a' is the real part and 'bi' is the imaginary part (i = √-1). These solutions always come in conjugate pairs: a + bi and a - bi.
How does the step-by-step solution work?
The solver breaks down the solution process into clear steps: identifying the equation type, converting to standard form, applying the appropriate solving method (substitution for linear, quadratic formula for quadratic), and verifying the solution. Each step includes the mathematical operation and explanation.
Can this solver help with homework?
Yes! The equation solver is perfect for checking homework answers and understanding solving methods. The step-by-step explanations help you learn the process, not just get the answer. Use it to verify your work and understand where you might have made mistakes.
How accurate are the solutions?
Solutions are computed in double-precision floating point and can be displayed with up to 10 decimal places. Every real solution is automatically verified by substituting it back into both sides of the original equation, and that verification step is shown in the step-by-step solution. Complex solutions are checked the same way using complex arithmetic.