Algebra Calculator: How to Solve Linear and Quadratic Equations Step by Step

Algebra is the foundation of nearly all advanced mathematics — from engineering and physics to economics and data science. The good news: most real-world algebra problems reduce to two shapes — linear equations and quadratic equations — and both follow a small set of steps you can learn once and reuse forever. This guide walks through solving each type by hand, then shows how to verify your answer instantly with the Algebra Calculator.

Solving Linear Equations

A linear equation has the form ax + b = c — the variable appears only to the first power. The strategy is always the same: isolate x by undoing operations in reverse order.

Example: Solve 3x + 7 = 22

1. Subtract 7 from both sides: 3x = 15
2. Divide both sides by 3: x = 5
3. Check: 3(5) + 7 = 15 + 7 = 22 ✓

When the variable appears on both sides (e.g. 5x - 3 = 2x + 9), first move all variable terms to one side and constants to the other, then solve. A quick way to sanity-check any linear equation: plug your answer back in — if both sides match, you’re done.

Solving Quadratic Equations

Quadratics have the form ax² + bx + c = 0 and can have up to two solutions. Three methods cover almost every case:

• Factoring — fastest when the equation factors cleanly (e.g. x² - 5x + 6 = 0 → (x-2)(x-3) = 0 → x = 2 or x = 3)
• Quadratic formula — always works: x = (-b ± √(b² - 4ac)) / 2a
• Completing the square — useful for deriving the vertex of a parabola

Example using the quadratic formula: Solve 2x² - 4x - 6 = 0. Here a=2, b=-4, c=-6. Discriminant: b² - 4ac = 16 + 48 = 64. Then x = (4 ± 8) / 4, giving x = 3 or x = -1.

The Discriminant Tells You How Many Solutions

Before solving, compute the discriminant D = b² - 4ac:

Discriminant	Solutions	Graph
D > 0	Two real solutions	Parabola crosses the x-axis twice
D = 0	One real solution	Parabola touches the x-axis once
D < 0	No real solutions	Parabola never touches the x-axis

Systems of Equations

When you have two equations with two unknowns, use substitution (solve one equation for a variable, plug into the other) or elimination (add or subtract equations to cancel a variable). Both methods give the same intersection point of two lines.

Verify Your Work with the Algebra Calculator

Hand-solving is great for learning, but for homework, exam prep, or a quick work calculation, the Algebra Calculator solves linear equations, quadratics, and systems instantly — with the step-by-step breakdown shown so you can compare it against your own work. Enter your equation, check the solution, and if your answer differs, the steps will show you exactly where the paths diverge. Try it now: solve a few practice problems by hand first, then verify with the calculator to build speed and confidence.