Algebra Solver

How to Solve Quadratic Equations — The Quadratic Formula

Any equation in the form $ax^2 + bx + c = 0$ can be solved with the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

The ± means there are generally two solutions — one using + and one using −. This formula works for every quadratic equation, regardless of whether it factors neatly, which is why it's the most reliable general-purpose method.

Worked Example: Solving $x^2 - 5x + 6 = 0$

Here $a=1$, $b=-5$, $c=6$. The discriminant is $(-5)^2 - 4(1)(6) = 25 - 24 = 1$ — positive, so expect two real roots. Plugging into the formula: $x = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2}$, giving $x = 3$ or $x = 2$. Checking by factoring confirms it: $x^2 - 5x + 6 = (x-3)(x-2)$, which is zero exactly when $x=3$ or $x=2$.

The Discriminant — What It Tells You About the Roots

The expression under the square root, $b^2 - 4ac$, is called the discriminant, and its sign alone tells you what kind of roots to expect before you finish solving. A positive discriminant means two distinct real roots — the parabola crosses the x-axis twice. A discriminant of exactly zero means one repeated real root — the parabola just touches the x-axis at its vertex. A negative discriminant means the roots are complex numbers — the parabola never touches the x-axis at all.

Systems of Equations — Substitution vs. Elimination vs. Cramer's Rule

Substitution solves one equation for a variable and plugs it into the other — intuitive but can get messy with awkward coefficients. Elimination adds or subtracts multiples of the equations to cancel out a variable — often faster by hand. Cramer's rule, which this calculator uses, expresses the solution directly in terms of determinants of the coefficient matrix — it's especially clean for exactly two equations and generalizes well to larger systems. The rule is named after Swiss mathematician Gabriel Cramer, who published it in 1750, though the Scottish mathematician Colin Maclaurin had described an equivalent method a few years earlier.

Complex Roots Explained

When a quadratic's discriminant is negative, the square root of a negative number appears — which isn't a real number. Mathematicians define i = √-1 to handle this, giving roots in the form a ± bi, where a is the real part and b is the imaginary part. These roots don't correspond to any x-intercept on a standard graph, but they're valid solutions in the complex number system and matter in fields like electrical engineering and signal processing.

Solving Larger Systems with Gaussian Elimination

Cramer's rule works cleanly for two equations, but it gets unwieldy fast as the number of unknowns grows. Despite its name, elimination-based solving predates Carl Friedrich Gauss by centuries — a version appears in the ancient Chinese text "The Nine Chapters on the Mathematical Art" — but the method became known as Gaussian elimination after 19th-century Western mathematicians adopted the systematic notation Gauss had used for it. Gaussian elimination scales better: it uses row operations — swapping rows, and subtracting a multiple of one row from another — to turn the augmented matrix into an upper-triangular form where the last row has only one unknown left. From there, back-substitution works upward, solving one variable at a time and plugging each result into the row above it. The System (N) mode on this page walks through every row operation for up to 6 equations, exactly as you'd do it by hand.

A system can land in one of three outcomes: a single unique solution (the usual case), infinitely many solutions (when one equation is a combination of the others, leaving a free variable), or no solution at all (when the equations contradict each other, like requiring x + y to equal both 5 and 8). This calculator detects all three automatically from the row-reduced matrix.

A Brief History of Algebra

The word "algebra" comes from "al-jabr," part of the title of a 9th-century treatise by the Persian mathematician Muhammad ibn Musa al-Khwarizmi, whose systematic methods for solving linear and quadratic equations laid much of the groundwork for the field — his name is also the root of the word "algorithm." Earlier civilizations, including the Babylonians (who had methods equivalent to the quadratic formula nearly 4,000 years ago, though expressed geometrically rather than symbolically) and the ancient Greeks and Indians, developed their own techniques for solving equations, but al-Khwarizmi's work was distinctive for treating equation-solving as a general, reusable method rather than a collection of specific problems. The symbolic notation familiar today — using letters for unknowns and clean symbols for operations — developed much later, largely through the work of French mathematician François Viète in the late 1500s and further refined by René Descartes in the 1600s.

Common Algebra Mistakes

Forgetting the ± when taking a square root during a derivation is one of the most common errors, silently discarding one of two valid solutions. Sign errors when distributing a negative across parentheses — turning $-(x - 3)$ into $-x - 3$ instead of the correct $-x + 3$ — are another frequent source of wrong answers. When solving systems, mixing up which equation a substituted expression came from, or forgetting to check a solution in the original (not just the simplified) equation, can also silently introduce errors that a quick verification step would catch.

Algebra Terms You Should Know

Coefficient — the numerical factor multiplying a variable, such as the 3 in $3x$.

Root (or Zero) — a value of the variable that makes the equation equal zero; for a quadratic, these are the x-intercepts of its graph.

Discriminant — the expression $b^2 - 4ac$ under the square root in the quadratic formula, whose sign predicts the number and type of roots.

Imaginary Unit — denoted $i$, defined as the square root of $-1$, used to express roots that don't correspond to a real number.

Augmented Matrix — a matrix combining a system's coefficients and constants into a single grid, used to solve systems of equations via row operations.