Graphing Quadratic Inequalities Calculator
Solve graphing quadratic inequalities problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Calculator
Adjust values & calculate{x | 1 < x < 2}
The parabola opens upward. The solution is between the roots where the parabola is below the x-axis.
Test Regions
Formula
Find roots using the quadratic formula, then determine which intervals satisfy the inequality based on the parabola direction (sign of a) and the inequality operator.
Last reviewed: December 2025
Worked Examples
Example 1: Solve x^2 - 3x + 2 < 0
Example 2: Solve -2x^2 + 8x - 6 >= 0
Background & Theory
The Graphing Quadratic Inequalities Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Graphing Quadratic Inequalities Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
ax^2 + bx + c [<, <=, >, >=] 0
Find roots using the quadratic formula, then determine which intervals satisfy the inequality based on the parabola direction (sign of a) and the inequality operator.
Worked Examples
Example 1: Solve x^2 - 3x + 2 < 0
Problem: Find all x-values where x^2 - 3x + 2 is less than zero.
Solution: Factor: x^2 - 3x + 2 = (x - 1)(x - 2)\nRoots: x = 1 and x = 2\nSign analysis:\n x < 1: both factors negative, product positive\n 1 < x < 2: (x-1) positive, (x-2) negative, product negative\n x > 2: both factors positive, product positive\nWe want < 0, so the solution is the interval where the product is negative.\nSolution: 1 < x < 2
Result: Solution: (1, 2)
Example 2: Solve -2x^2 + 8x - 6 >= 0
Problem: Find all x-values where -2x^2 + 8x - 6 is greater than or equal to zero.
Solution: Factor out -2: -2(x^2 - 4x + 3) = -2(x - 1)(x - 3)\nRoots: x = 1 and x = 3\nSince a = -2 < 0, parabola opens downward.\nThe expression is non-negative BETWEEN the roots.\nSign analysis:\n x < 1: product negative\n 1 <= x <= 3: product non-negative\n x > 3: product negative\nSolution: 1 <= x <= 3
Result: Solution: [1, 3]
Frequently Asked Questions
What is a quadratic inequality and how does it differ from a quadratic equation?
A quadratic inequality is a mathematical statement that compares a quadratic expression to zero using an inequality sign (<, <=, >, or >=) rather than an equals sign. While a quadratic equation ax^2 + bx + c = 0 asks for the specific x-values where the parabola crosses the x-axis, a quadratic inequality asks for all x-values where the parabola is above or below the x-axis. The solution to a quadratic equation is typically two points (the roots), while the solution to a quadratic inequality is typically one or more intervals on the number line. For example, x^2 - 4 = 0 has solutions x = -2 and x = 2, but x^2 - 4 < 0 has the solution interval (-2, 2), representing all x-values where the parabola is below the axis.
How do you solve a quadratic inequality step by step?
Solving a quadratic inequality follows a systematic process. First, rewrite the inequality with zero on one side and the quadratic expression on the other. Second, find the roots of the corresponding equation ax^2 + bx + c = 0 using factoring, completing the square, or the quadratic formula. Third, plot the roots on a number line, dividing it into regions (typically two or three intervals). Fourth, test one point from each region by substituting it into the original inequality. Fifth, determine which regions satisfy the inequality. Finally, write the solution using interval notation, remembering to use parentheses for strict inequalities and brackets for non-strict ones. This sign-analysis method works because quadratic functions only change sign at their roots.
How do you graph a quadratic inequality on a coordinate plane?
Graphing a quadratic inequality on a coordinate plane involves several steps. First, graph the corresponding parabola y = ax^2 + bx + c. Use a solid curve for non-strict inequalities (<= or >=) since points on the parabola are included in the solution, and a dashed curve for strict inequalities (< or >) since boundary points are excluded. Second, determine which region to shade. For y < ax^2 + bx + c or y <= ..., shade below the parabola. For y > ... or y >= ..., shade above the parabola. The shaded region represents all (x, y) coordinate pairs that satisfy the inequality. Test a point like (0, 0) to verify you shaded the correct region.
What is the test point method for quadratic inequalities?
The test point method involves selecting a sample value from each region created by the roots and substituting it into the quadratic expression to determine the sign of the expression in that region. After finding the roots, the number line is divided into intervals. Pick any convenient number from each interval (often integers or zero are easiest to compute). Substitute each test point into the original quadratic expression and determine whether the result is positive or negative. If the result satisfies the inequality, the entire interval is part of the solution. This works because a continuous quadratic expression can only change sign at its roots, so the sign is constant throughout each interval between consecutive roots.
How do boundary points work with strict vs non-strict inequalities?
Boundary points are the roots of the corresponding quadratic equation, and their inclusion or exclusion in the solution set depends on whether the inequality is strict or non-strict. For strict inequalities (< and >), the boundary points are NOT included in the solution because the expression equals zero at these points, and zero is not strictly less than or greater than zero. For non-strict inequalities (<= and >=), the boundary points ARE included because zero satisfies both <= 0 and >= 0. In interval notation, excluded boundaries are marked with parentheses and included boundaries with brackets. On a number line graph, excluded points get open circles and included points get closed (filled) circles.
Can quadratic inequalities have no solution or infinitely many solutions?
Yes, both cases are possible. A quadratic inequality has no solution (the empty set) when the condition can never be satisfied. For example, x^2 + 1 < 0 has no solution because x^2 + 1 is always positive (minimum value is 1). A quadratic inequality has all real numbers as solutions when the condition is always satisfied. For example, x^2 + 1 > 0 is true for every real number. These special cases arise when the discriminant is negative (no real roots) or when the discriminant is zero and the inequality is non-strict. A single-point solution set can also occur, such as x^2 <= 0, which is only satisfied at x = 0.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy