Understanding Linear Quadratic Systems
A linear quadratic system joins a line and a parabola. The solution points are the places where both equations share the same x and y values. This calculator follows the algebraic method. It sets the line equal to the quadratic expression. Then it solves the resulting quadratic equation.
Why substitution works
The core idea is substitution. For a line written as y = mx + d, replace y in the quadratic equation. The new equation becomes ax² + (b - m)x + (c - d) = 0. The discriminant decides the number of real intersections. A positive value gives two points. A zero value gives one tangent point. A negative value gives no real point.
Detailed classroom checking
This tool is built for detailed classroom work. It accepts decimal coefficients, negative values, and vertical line cases. It also shows the adjusted coefficients used in the final equation. Each root is substituted back into the selected line rule. That gives the matching y value. The calculator also reports the vertex of the parabola. This helps compare the curve shape with the line.
Rounding and exact meaning
Use the decimal option when you need rounded answers. Use more decimal places for graphing or engineering notes. Use fewer places when the result is for quick homework checking. The exact radical form is explained by the discriminant. When the square root is not a clean number, the decimal roots are still useful.
Exporting and checking signs
The example table shows common cases. It includes secant, tangent, and non intersecting systems. These examples help you test the calculator before entering your own values. The CSV export is useful for spreadsheets. The PDF export is useful for sharing a clean report.
Always review the input signs. A small sign change can move a line across the parabola. That can change two intersections into one or none. For best results, write both equations clearly first. Then enter each coefficient in the matching field.
Why algebra matters
Algebra also gives stronger evidence than a graph alone. A graph may hide close intersections. It may also suggest a point that is only approximate. The formula shows whether answers are exact or rounded. That makes the method reliable for tests, notes, and reports. It also explains why each answer appears. Students can compare the table, formula, and result before completing their work carefully.