Solve linear and quadratic equations, factor polynomials, expand with FOIL, and solve 2×2 systems — all with step-by-step solutions
A linear equation is any equation where the variable appears with a power of 1 — no squares, cubes, or higher powers. The standard form ax + b = c has a single solution for x when the coefficient a is non-zero: subtract b from both sides to get ax = c − b, then divide both sides by a to isolate x = (c − b) / a. This two-step process — moving constants to one side and dividing by the coefficient — is the foundation of all algebraic manipulation. When a = 0, the equation either has infinite solutions (if b = c, making it always true) or no solution (if b ≠ c, making it a contradiction). Understanding these edge cases is as important as solving the typical case.
Real-world applications of linear equations are everywhere. If you earn $22 per hour and have already worked enough to earn $66, how many total hours would give you $154 in earnings? Set up 22h = 154 − 66... actually, more naturally: 22h + 66 = 154, so h = (154 − 66) / 22 = 88 / 22 = 4 more hours. Every proportional relationship — unit pricing, speed-distance-time, temperature conversions — can be modeled as a linear equation. The ability to isolate any single variable in a linear relationship is the practical skill that algebra unlocks.
A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0. Unlike linear equations which have exactly one solution, quadratics can have two distinct real solutions, one repeated solution, or no real solutions at all — determined entirely by the discriminant Δ = b² − 4ac. When Δ > 0, the square root is real and produces two distinct solutions. When Δ = 0, the ± term vanishes and both solutions collapse to x = −b / 2a. When Δ < 0, the square root of a negative number enters the picture, and solutions exist only as complex numbers involving the imaginary unit i = √−1.
The quadratic formula works for every quadratic equation without exception — it's derived by completing the square on the general form ax² + bx + c = 0 and produces exact solutions. For x² − 5x + 6 = 0: a = 1, b = −5, c = 6, Δ = 25 − 24 = 1, x = (5 ± 1) / 2, giving x = 3 and x = 2. You can verify both: 3² − 5(3) + 6 = 9 − 15 + 6 = 0 ✓ and 2² − 5(2) + 6 = 4 − 10 + 6 = 0 ✓. Always verify by substituting solutions back into the original equation — this is the gold standard of checking algebraic work.
Factoring reverses the multiplication process to express a polynomial as a product of simpler expressions. For ax² + bx + c with a = 1, the strategy is finding two numbers that multiply to c and add to b — then writing (x + p)(x + q) where pq = c and p + q = b. For x² + 5x + 6: find two numbers multiplying to 6 and adding to 5. The pair (2, 3) works: 2 × 3 = 6 and 2 + 3 = 5, so x² + 5x + 6 = (x + 2)(x + 3). When a ≠ 1, the process requires the AC method: find two numbers multiplying to a × c and adding to b, split the middle term, then factor by grouping.
Factoring connects directly to solving: if (x + 2)(x + 3) = 0, then by the Zero Product Property, either x + 2 = 0 or x + 3 = 0, giving x = −2 or x = −3. This is the fastest way to solve factorable quadratics — faster than the quadratic formula. The three most important patterns to memorize are the perfect square trinomial (a + b)² = a² + 2ab + b², the difference of squares a² − b² = (a + b)(a − b), and the sum/difference of cubes. Recognizing these patterns on sight eliminates the need for formula application and dramatically speeds up algebraic work.
FOIL stands for First, Outer, Inner, Last — the four products that arise when multiplying two binomials (ax + b)(cx + d). First: multiply the first terms of each binomial → ac·x². Outer: multiply the outer terms → ad·x. Inner: multiply the inner terms → bc·x. Last: multiply the last terms → bd. Combine the two middle terms (ad + bc)x to get the final trinomial: acx² + (ad + bc)x + bd. For (2x + 3)(3x + 4): First = 6x², Outer = 8x, Inner = 9x, Last = 12, combined = 6x² + 17x + 12.
FOIL is not a separate rule — it's just the distributive property applied twice. (ax + b)(cx + d) = ax(cx + d) + b(cx + d) = acx² + adx + bcx + bd. Understanding this derivation means you can extend the approach to larger polynomials: (a + b + c)(d + e) requires distributing each term of the first over each term of the second, producing six products rather than four. Recognizing FOIL as applied distribution also makes it easier to verify expansions and to reverse the process (factoring) when given the expanded trinomial.
A system of two linear equations with two unknowns represents two lines in a plane. The solution is the point where the lines intersect — the (x, y) pair satisfying both equations simultaneously. Three scenarios are possible: the lines intersect at exactly one point (unique solution), the lines are parallel and never meet (no solution), or the lines are identical (infinite solutions). Algebraically, these correspond to a non-zero determinant, a determinant of zero with inconsistent constants, and a determinant of zero with consistent constants respectively.
This calculator uses elimination (also called Cramer's Rule for 2×2 systems) to find the solution. Given equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the determinant is D = a₁b₂ − a₂b₁. When D ≠ 0, the unique solution is x = (c₁b₂ − c₂b₁) / D and y = (a₁c₂ − a₂c₁) / D. Substitution is the alternative approach — solve one equation for one variable, then substitute into the other. Both methods always yield the same answer; elimination tends to be faster when coefficients are arranged favorably, while substitution shines when one equation is already solved for a variable.
Choosing the right method saves significant time. For linear equations, there's really only one method — isolate the variable by inverse operations — but the order matters: clear fractions first by multiplying by the LCD, then eliminate parentheses by distributing, collect like terms on each side, and finally isolate the variable. For quadratics, factoring is fastest when the discriminant is a perfect square and the factors are integers. The quadratic formula works universally but involves more arithmetic. Completing the square is most useful for converting to vertex form y = a(x − h)² + k for graphing purposes.
For systems of equations, elimination works best when coefficients of one variable are equal or opposites (multiply-to-match is needed otherwise). Substitution is best when one equation has an isolated variable like y = 2x + 3. Graphing is useful for visualization and for estimating solutions when exact values aren't required. Matrix methods (row reduction, Gaussian elimination) generalize to larger systems of 3, 4, or more equations and variables, using the same logic but applied systematically to arrays of coefficients.
The most frequent algebra error is incorrect distribution of negatives. The expression −(2x + 3) equals −2x − 3, not −2x + 3 — the negative sign must distribute to every term inside the parentheses, not just the first. Similarly, −(x − 4) = −x + 4, because subtracting a negative yields a positive. This error is especially common when moving terms across the equals sign or when expanding expressions like (x − 3)²: the correct expansion is x² − 6x + 9, not x² − 6x − 9 or x² + 9.
Another extremely common mistake is dividing by a variable before checking whether it could equal zero. If you have x(x − 3) = 0, dividing both sides by x would give x − 3 = 0, so x = 3 — but this loses the solution x = 0 entirely. The Zero Product Property is the correct approach: if a product equals zero, at least one factor must equal zero. A third common error is misapplying the square root operation: if x² = 9, then x = ±3 (both +3 and −3 are solutions), not just x = 3. Forgetting the ± when taking square roots of both sides is one of the most common sources of missing solutions in quadratic problems.