Algebra Solver

How to Use This Algebra Solver

Algebra by Student Level

Using This Tool for Algebra Assignments

What Is Algebra?

What is algebra in math?

Algebra is the branch of mathematics that uses letters and symbols to represent unknown numbers. Instead of asking "what number plus 5 equals 11?" you write x + 5 = 11 and solve for x.

What are variables and expressions?

A variable (most often x, y, or z) stands in for an unknown number. An expression is a combination of variables and numbers, like 3x + 5. An equation adds an equals sign: 3x + 5 = 11.

Why is algebra important?

Algebra teaches step-by-step logical thinking and underpins every STEM subject. It appears on every standardized test from GCSE and SAT to AP and university entrance exams.

Types of Algebraic Equations

How to Solve Linear and Quadratic Equations

How do I solve for x?

Use inverse operations — undo each operation applied to x. For 2x + 3 = 7: subtract 3 from both sides → 2x = 4, then divide by 2 → x = 2.

What is the quadratic formula?

The discriminant Δ = b² − 4ac tells you everything before you finish: Δ > 0 gives two real roots, Δ = 0 gives one repeated root, Δ < 0 gives two complex roots.

Higher-Degree Polynomial Equations

What is a cubic equation?

A cubic equation has degree 3: ax³ + bx² + cx + d = 0. It can have one, two, or three real roots. The graph is an S-shaped curve that always crosses the x-axis at least once.

What is a quartic equation?

A quartic equation has degree 4: ax⁴ + bx³ + cx² + dx + e = 0. It can have zero, two, or four real roots. Some quartic equations can be solved by substituting u = x² to turn them into a quadratic.

How does the solver find roots of cubic and quartic equations?

This tool uses the Newton-Raphson method combined with interval scanning. It evaluates the polynomial across a wide range, detects sign changes (where a root must exist), then refines each root with rapid Newton-Raphson iteration.

What is the Rational Root Theorem?

For a polynomial with integer coefficients, any rational root p/q must have p as a factor of the constant term and q as a factor of the leading coefficient. This lets you test a short list of candidates before using the quadratic formula or numerical methods on the remaining factor.

What is synthetic division?

Synthetic division is a shortcut for dividing a polynomial by (x − r). Once you find one root r, you divide the polynomial by (x − r) to get a lower-degree polynomial, then solve that. This "factor and reduce" process is the standard method for higher-degree polynomials by hand.

Solving Inequalities in Algebra

What is an algebraic inequality?

An inequality uses <, >, ≤, or ≥ instead of =. The solution is not a single point but a range of values, written as an interval like (−∞, 3] or (2, +∞).

How do I solve a linear inequality?

Treat it like a linear equation — move terms, combine, divide. The one critical rule: when you multiply or divide both sides by a negative number, you must flip the inequality sign. For example: −2x > 6 → divide by −2 and flip → x < −3.

How do I solve a quadratic inequality like x² − 4 > 0?

First find the roots: x² − 4 = 0 → x = ±2. These split the number line into three intervals. Test a point in each interval to see where the inequality holds. For x² − 4 > 0 (upward parabola): x < −2 or x > 2.

What is interval notation?

Interval notation writes the solution range as (a, b) for strict inequality (open endpoints) or [a, b] for ≤ or ≥ (closed endpoints). A parenthesis means that endpoint is NOT included; a bracket means it IS included. Infinity always uses a parenthesis: (−∞, 3].

What is the sign chart method for inequalities?

After finding the roots of f(x), plot them on a number line. Pick a test point in each interval and evaluate f(x). If the result is positive, f(x) > 0 in that whole interval; if negative, f(x) < 0. Shade the intervals that satisfy your original inequality.

Rational and Radical Equations

What is a rational equation?

A rational equation has a variable inside a fraction (denominator). For example: 2/(x−1) + 3/x = 5. The strategy is to find the LCD (least common denominator), multiply every term by it to clear the fractions, then solve the resulting polynomial equation.

What is an extraneous solution?

When solving rational equations, multiplying by the LCD can sometimes introduce solutions that make a denominator zero. These are called extraneous solutions — they satisfy the transformed equation but not the original. Always substitute your answer back to check.

What is a radical equation?

A radical equation has a variable under a square root, cube root, or other root. For example: √(x + 5) = 4. The strategy is to isolate the radical, then raise both sides to the power of the root index (squaring for square roots). Then solve the resulting polynomial.

How do I solve √(x + 5) = 4?

Square both sides: x + 5 = 16. Subtract 5: x = 11. Always verify: √(11 + 5) = √16 = 4 ✓. This solver handles basic radical equations of the form √(x + a) = b — for complex nested radicals, use the AI Math Solver.

How do I solve 3/x = 6?

Multiply both sides by x: 3 = 6x. Divide by 6: x = 1/2. Verify: 3/(1/2) = 6 ✓. Enter 3/x = 6 into the solver for the full working. More complex rational equations with multiple fractions are best handled by the AI Math Solver.

Exponential and Logarithmic Equations

What is an exponential equation?

An exponential equation has the variable in the exponent position, like 2^x = 16 or e^x = 7. These cannot be solved by ordinary algebra. You must use logarithms to "bring down" the exponent.

How do I solve 2^x = 16?

Recognize that 16 = 2⁴, so x = 4. For less obvious cases, take the log of both sides: x = log(16) / log(2). This tool handles exponential equations in the form a^x = b and e^x = b with full step-by-step working.

What is a logarithmic equation?

A logarithmic equation has the variable inside a log function, like log(x) = 3 or ln(x) = 2. The inverse of log₁₀(x) = b is x = 10^b. The inverse of ln(x) = b is x = e^b.

How do I solve log(x) = 3?

Rewrite in exponential form: x = 10³ = 1000. This tool supports log(x) = b (base 10) and ln(x) = b (natural log). Enter exactly as written: log(x) = 3 or ln(x) = 2.

What is the change-of-base formula?

To compute log₅(125) on a calculator, use: log(125) / log(5) = 3. This converts any logarithm to base 10 or natural log. It is especially useful when solving equations like 5^x = 125 → x = log(125)/log(5).

Why is e important in algebra?

Euler's number e ≈ 2.718 is the base of natural logarithms. It appears naturally in compound interest, population growth, radioactive decay, and many calculus problems. The natural log ln(x) is the inverse of e^x.

Matrix Algebra and Systems of Equations

What is a system of 3 equations with 3 unknowns?

Three equations, three unknowns (x, y, z). The unique solution is the point (x, y, z) where all three planes intersect. Enter all three equations separated by semicolons: x+y+z=6; 2x-y+z=3; x+2y-z=2.

What is Gaussian elimination?

Gaussian elimination (row reduction) transforms the augmented matrix [A|b] into an upper triangular form using row operations. Then back-substitution gives the solution. This tool applies partial pivoting for numerical stability.

What is Cramer's Rule?

Cramer's Rule gives a direct formula for each variable using determinants. For a 2×2 system: x = (c₁b₂ − c₂b₁) / D where D = a₁b₂ − a₂b₁. It is elegant for 2×2 systems. Gaussian elimination scales better for 3×3 and larger.

What is a determinant?

The determinant of a 2×2 matrix [[a, b], [c, d]] is ad − bc. A non-zero determinant means the system has a unique solution. A zero determinant means the lines (or planes) are parallel or identical — no unique solution exists.

What is a matrix in algebra?

A matrix is a rectangular array of numbers arranged in rows and columns. In algebra, matrices are used to represent and solve systems of equations efficiently, especially when the number of variables is large. Operations like addition, multiplication, and finding the inverse are core to linear algebra.

Graphing Algebraic Equations

What does the graph show in this solver?

After solving, the tool draws y = f(x) where f(x) = (left side) − (right side). Where the curve crosses y = 0 (the x-axis) are the roots of the equation, marked with red dots. This visual confirms your algebraic answer.

How do I read a parabola graph for a quadratic?

A parabola opens upward when a > 0 and downward when a < 0. Where it crosses the x-axis are the real roots. If it touches but does not cross, there is one repeated root. If it never crosses, all roots are complex.

How does the graph help with inequalities?

For an inequality like x² − 4 > 0, the graph shades the x-regions where the curve is above the x-axis. This gives an immediate visual answer. Blue shading shows where the inequality holds — read the x-interval from the graph edges.

What is the Cartesian coordinate plane?

The Cartesian plane has two perpendicular axes: the horizontal x-axis and the vertical y-axis. Any point is identified by its (x, y) coordinates. Functions are graphed by plotting (x, f(x)) for many values of x and connecting the points into a curve.

How do I interpret where a line or curve crosses the x-axis?

At any x-intercept, y = 0. Since our graph plots y = f(x) = left − right, an x-intercept means f(x) = 0, which is exactly where the original equation is satisfied. So every x-intercept IS a solution to the equation.