Graphing Quadratic Inequalities Calculator
Solve graphing quadratic inequalities problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
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) Roots: x = 1 and x = 2 Sign analysis: x < 1: both factors negative, product positive 1 < x < 2: (x-1) positive, (x-2) negative, product negative x > 2: both factors positive, product positive We want < 0, so the solution is the interval where the product is negative. Solution: 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) Roots: x = 1 and x = 3 Since a = -2 < 0, parabola opens downward. The expression is non-negative BETWEEN the roots. Sign analysis: x < 1: product negative 1 <= x <= 3: product non-negative x > 3: product negative Solution: 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 does the sign of the leading coefficient affect the solution?
The leading coefficient a determines whether the parabola opens upward (a > 0) or downward (a < 0), which fundamentally affects which regions satisfy the inequality. When a > 0 (opens up), the parabola is negative (below the x-axis) between the roots and positive outside the roots. When a < 0 (opens down), the parabola is positive between the roots and negative outside. For a less-than inequality with a > 0, the solution is the interval between the roots. For a less-than inequality with a < 0, the solution is the two intervals outside the roots. This pattern reverses for greater-than inequalities. Understanding this relationship eliminates the need for testing points in each region once you know the sign of a.
What happens when the discriminant is negative?
When the discriminant b^2 - 4ac is negative, the quadratic equation has no real roots, meaning the parabola never crosses or touches the x-axis. In this case, the parabola is entirely above the x-axis (if a > 0) or entirely below it (if a < 0). For the inequality, this means either all real numbers are solutions or no real numbers are solutions, depending on the direction of the inequality. If a > 0 and the inequality is > 0 or >= 0, the solution is all real numbers. If a > 0 and the inequality is < 0 or <= 0, there is no solution. The situation reverses when a < 0. This makes negative discriminant cases the simplest to solve once you determine the sign of a.
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.
How do quadratic inequalities apply to real-world problems?
Quadratic inequalities model many real-world scenarios involving optimization and constraints. In projectile motion, the inequality -16t^2 + 64t > 48 determines the time interval during which a ball is above 48 feet. In business, a profit function P(x) = -5x^2 + 200x - 1500 > 0 determines the range of production quantities that generate positive profit. In engineering, material stress constraints involve quadratic inequalities to ensure safety factors are met. In agriculture, crop yield models determine optimal fertilizer ranges. These applications demonstrate why solving quadratic inequalities is not merely an academic exercise but a practical tool for decision-making in fields involving parabolic relationships.
How does factoring help solve quadratic inequalities?
Factoring transforms the quadratic inequality into a product of linear factors, making sign analysis straightforward. If ax^2 + bx + c factors as a(x - r1)(x - r2), then the sign of the expression depends on the signs of each factor in different regions. At any point x, the factor (x - r1) is negative when x < r1 and positive when x > r1, and similarly for (x - r2). By creating a sign chart showing the sign of each factor in each interval, you can determine the sign of the product. A product of two negatives is positive; a product of a positive and a negative is negative. This method gives the solution without needing to evaluate test points numerically and provides deeper insight into why the solution takes its particular form.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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