Solve linear equations, quadratic equations, and systems of linear equations for free. View discriminant, step-by-step solutions, and complex roots.
The discriminant (D) is calculated as D = b² - 4ac and determines the type of roots. If D > 0, there are two distinct real roots; if D = 0, there is one repeated root; if D < 0, there are two complex conjugate roots. The discriminant directly relates to the number of points where the parabola intersects the x-axis.
Complex roots consist of a real part and an imaginary part, expressed as a + bi, where i represents √(-1). Complex roots always appear in conjugate pairs. Graphically, this means the parabola does not intersect the x-axis.
This solver uses Cramer's Rule, which calculates each variable directly using determinants. If the determinant is 0, the system has no solution (inconsistent) or infinitely many solutions (dependent).
In ax + b = 0, if a = 0, the equation becomes 0x + b = 0, which is not an equation in x. If b is also 0, every real number is a solution (identity). If b ≠ 0, there is no solution. Therefore, a must be non-zero for a valid linear equation.
An equation is an equality containing unknowns and is one of the most fundamental tools in mathematics. The linear equation ax + b = 0 is equivalent to finding the x-intercept of a line and is solved simply as x = -b/a. The quadratic equation ax² + bx + c = 0 is solved using the quadratic formula x = (-b ± √(b² - 4ac)) / 2a, yielding real or complex roots depending on the discriminant D = b² - 4ac. Equation solving is essential in physics (equations of motion), economics (equilibrium price), engineering (circuit analysis), and many other fields.
A system of equations involves finding values that simultaneously satisfy two or more equations. A 2x2 system can be solved using Cramer's Rule or Gaussian elimination, which geometrically corresponds to finding the intersection of two lines. If the determinant is 0, the lines are parallel (no solution) or coincident (infinite solutions). The ability to solve equations is foundational to all quantitative fields including science, engineering, and data analysis, making it important to understand the solution process thoroughly.