Equation Solver Calculator

Linear Equations

A linear equation is a first-degree equation that can be written in the form ax + b = c, where a, b, and c are constants and a ≠ 0. The term "linear" refers to the fact that the graph of such an equation is a straight line.

To solve a linear equation:

1. Simplify both sides of the equation by combining like terms
2. Move all variable terms to one side and all constant terms to the other side
3. Divide both sides by the coefficient of the variable to isolate the variable

For example, to solve 2x + 5 = 15:

• Subtract 5 from both sides: 2x = 10
• Divide both sides by 2: x = 5

Linear equations have exactly one solution, unless they represent a contradiction (no solution) or an identity (infinitely many solutions).

Applications of linear equations include calculating distances, costs, temperatures, and countless other real-world scenarios where quantities are directly proportional.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

There are several methods to solve quadratic equations:

1. Factoring Method

If the quadratic expression can be factored, we can write it as a product of two linear factors: ax² + bx + c = (px + q)(rx + s) = 0.

By the zero product property, either px + q = 0 or rx + s = 0, which gives us the solutions x = -q/p or x = -s/r.

2. Quadratic Formula

The solutions to a quadratic equation ax² + bx + c = 0 are given by:

x = (-b ± √(b² - 4ac)) / (2a)

The term b² - 4ac is called the discriminant. Its value determines the number and nature of the solutions:

• If b² - 4ac > 0, there are two distinct real solutions
• If b² - 4ac = 0, there is exactly one real solution (a repeated root)
• If b² - 4ac < 0, there are two complex conjugate solutions

3. Completing the Square

This method involves rewriting the quadratic expression in the form a(x + d)² + e, which makes it easier to find the solutions.

For example, to solve x² - 3x - 4 = 0:

• Using the quadratic formula: a = 1, b = -3, c = -4
• x = (3 ± √(9 + 16)) / 2 = (3 ± √25) / 2 = (3 ± 5) / 2
• x = 4 or x = -1

Quadratic equations appear in many practical applications, including problems involving area, projectile motion, optimization, and economics.

Polynomial Equations

A polynomial equation has the form a₀ + a₁x + a₂x² + ... + aₙxⁿ = 0, where n is a non-negative integer representing the degree of the polynomial and a₀, a₁, ..., aₙ are constants with aₙ ≠ 0.

Linear and quadratic equations are special cases of polynomial equations with degrees 1 and 2, respectively. Polynomial equations of degree 3 or higher can be more challenging to solve and often require special techniques.

Methods for Solving Polynomial Equations:

1. Factoring: If the polynomial can be factored, we can apply the zero product property to find the solutions.
2. Rational Root Theorem: For polynomials with integer coefficients, any rational root p/q (in lowest form) must have p dividing the constant term and q dividing the leading coefficient.
3. Synthetic Division: A shorthand method for dividing a polynomial by a linear factor (x - r), useful for checking potential roots.
4. Numerical Methods: For polynomials that cannot be factored easily, numerical approaches like Newton's method can approximate the solutions.

For cubic equations (degree 3), there is a cubic formula similar to the quadratic formula, but it's much more complex. For degree 4, there's a quartic formula. The Abel-Ruffini theorem proves that there is no general algebraic formula for polynomial equations of degree 5 or higher.

For example, to solve x³ - 6x² + 11x - 6 = 0:

• Try to find one root using the rational root theorem. Potential rational roots are the factors of -6: ±1, ±2, ±3, ±6.
• Testing x = 1: 1³ - 6(1)² + 11(1) - 6 = 1 - 6 + 11 - 6 = 0 ✓
• So (x - 1) is a factor. Dividing by (x - 1) gives x² - 5x + 6.
• Factoring x² - 5x + 6 = (x - 2)(x - 3)
• Therefore, the solutions are x = 1, x = 2, and x = 3.

Polynomial equations arise in many areas, including physics, engineering, economics, and computer graphics.

Special Types of Equations