Solve and graph inequalities of the form ax² + bx + c < d, ≤ d, > d, ≥ d with instant visual results.
This interactive calculator solves and graphs quadratic inequalities of the form ax² + bx + c < d, ≤ d, > d, ≥ d. It provides instant interval solutions and a visual graph comparing the parabola with the horizontal line y = d.
Handles inequalities in the form:
Automatically rearranges the equation into standard quadratic form before solving.
The calculator plots:
This makes it easy to visually understand where the inequality is true.
The calculator computes the discriminant:
It determines whether there are two solutions, one double root, or no real solutions — and builds the correct interval accordingly.
As soon as you enter coefficients a, b, c, and d, the calculator:
1️⃣ You enter coefficients a, b, c, choose an inequality sign, and enter the value d.
2️⃣ The calculator rewrites the inequality:
ax² + bx + c − d ≷ 0
3️⃣ It calculates the discriminant:
Δ = b² − 4a(c − d)
4️⃣ Based on the value of Δ:
5️⃣ The solution interval depends on:
6️⃣ Finally, the graph visually shows where the parabola lies above or below the line y = d.
To solve a quadratic inequality of the form ax² + bx + c < d, ≤ d, > d, or ≥ d, we first rewrite it in standard form: ax² + bx + (c − d) ≷ 0. This step moves all terms to one side of the inequality so that the right-hand side becomes zero.
Standard Form: ax² + bx + (c − d) ≷ 0
Once in standard form, we calculate the discriminant (Δ) to determine the roots of the quadratic expression. The discriminant formula is:
Δ = b² − 4a(c − d)
Based on the value of Δ, we identify the nature of the roots and the solution intervals:
For inequalities, the solution interval depends on the parabola's direction (upward if a > 0, downward if a < 0) and the inequality sign:
Finally, the intervals are expressed using open ( ) or closed [ ] notation:
Example: x² − 4 > 0 → x < −2 OR x > 2
In this example, the parabola y = x² − 4 intersects y = 0 at x = −2 and x = 2. Since the inequality is greater than 0, the solution is the region above the x-axis, i.e., outside the roots.
Using these formulas, the calculator automatically computes the discriminant, determines the real roots, and displays the correct intervals. It also plots the parabola and horizontal line for visual understanding, making it easier to identify the solution graphically.
Suppose you enter:
x² − 4 > 0
The calculator solves:
x < −2 OR x > 2
The graph shows the parabola opening upward and lying above the x-axis outside the roots −2 and 2.
Everything you need to know about solving and graphing quadratic inequalities quickly and accurately.
It solves quadratic inequalities of the form ax² + bx + c < d, ≤ d, > d, ≥ d, and provides the solution intervals along with a visual graph comparing the parabola with the line y = d.
Yes! It is completely free to use with no registration or hidden charges required.
Yes, it supports all standard quadratic inequalities, including >, ≥, <, ≤, and handles cases where the parabola lies entirely above or below the line.
It uses the discriminant method (Δ = b² − 4a(c − d)) to find real roots, determines the intervals based on the inequality sign and whether the parabola opens upward or downward, and outputs the interval solution.
Yes, the calculator plots the quadratic parabola y = ax² + bx + c and the comparison line y = d to visually indicate the regions where the inequality holds true.
Yes, the calculator is fully responsive and works smoothly on smartphones, tablets, laptops, and desktops.
Use the "Clear All" button to reset input fields or the "Reload Calculator" button to reload the page entirely.
Absolutely! It is perfect for homework, quizzes, and exam preparation, providing instant, accurate results with clear visual and interval explanations.
It correctly handles open ( ) and closed [ ] intervals, infinite intervals (like (-∞, a) or [b, ∞)), and compound intervals when the inequality has multiple valid regions.
Yes, if the parabola never intersects the line y = d, the calculator clearly states "No solution" or "All real numbers" depending on the inequality type and parabola orientation.
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