mycalculators.app

Math Equation Solver

Solve mathematical equations instantly with detailed step-by-step solutions.

Enter Your Equation

Enter your equation using standard mathematical notation

Solution

x₁

3

Real number

x₂

2

Real number

Equation Type
quadratic
Solutions
2
Verified by substitution

Step-by-Step Solution

Step 1: Original equation
x^2 - 5x + 6 = 0
Step 2: Expand both sides
x² - 5x + 6 = 0

Distribute, expand products, and combine like terms on each side

Step 3: Move everything to the left side
x² - 5x + 6 = 0

Subtract the right side from both sides

Step 4: Identify coefficients
a = 1, b = -5, c = 6

Standard form: x² - 5x + 6 = 0

Step 5: Calculate the discriminant
Δ = b² - 4ac = (-5)² - 4(1)(6) = 1

Δ > 0 — two distinct real solutions

Step 6: Apply the quadratic formula
x = (-b ± √Δ) / (2a) = (5 ± √1) / 2
Step 7: Solutions
x₁ = 3, x₂ = 2
Step 8: Verify by substitution
x = 3: left side = 0, right side = 0 ✓ x = 2: left side = 0, right side = 0 ✓

Both solutions balance the original equation

Parabola Properties

Vertex
(2.5, -0.25)
Opens
Upward
Y-intercept
6
Axis of Symmetry
x = 2.5

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

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

  1. Simplify both sides by combining like terms
  2. Move all variable terms to one side
  3. Move all constant terms to the other side
  4. Divide by the coefficient of the variable
  5. Check your solution by substituting back

Example: Solve 3x + 7 = 2x + 15

3x + 7 = 2x + 15
3x - 2x = 15 - 7 (move terms)
x = 8 (simplify)

Quadratic Equation Methods

Factoring

Best when factors are integers

x² - 5x + 6 = 0
(x - 2)(x - 3) = 0
x = 2 or x = 3

Quadratic Formula

Works for any quadratic

x = (-b ± √(b²-4ac))/2a
Always gives exact answer
Handles complex roots

Completing Square

Useful for vertex form

x² + 6x + 5 = 0
(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

Example: x² - 5x + 6 = 0
Δ = 25 - 24 = 1
Roots: 3 and 2

Δ = 0

One repeated real root

Example: x² - 4x + 4 = 0
Δ = 16 - 16 = 0
Root: 2 (double)

Δ < 0

Two complex conjugate roots

Example: x² + 2x + 5 = 0
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
  1. Add 4: 5x = 20
  2. 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

What types of equations can this calculator solve?

Our math equation solver currently handles linear equations (ax + b = c) and quadratic equations (ax² + bx + c = 0). It provides real and complex solutions with step-by-step explanations. Support for polynomial, trigonometric, exponential, and systems of equations is coming soon.

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.

What's 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.

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.