Calculator Inputs
Example Data Table
| Case | Parabola | Line | Expected Result |
|---|---|---|---|
| Two intersections | y = x² - 3x + 2 | -x + y = 0 | Two real points |
| Tangent | y = x² | 0x + y = 0 | One tangent point |
| No real points | y = x² + 1 | 0x + y = 0 | Complex intersections |
| Vertical line | y = 2x² - x + 4 | 1x + 0y = 3 | One vertical intersection |
Formula Used
The calculator uses a parabola and a line:
Parabola: y = ax² + bx + c
Line: Lx + My = N
When M is not zero, the line becomes y = (N - Lx) / M.
Substitution gives: aMx² + (bM + L)x + (cM - N) = 0.
The discriminant is D = B² - 4AC. If D is positive, there are two real intersections. If D is zero, the line is tangent. If D is negative, there are complex intersections or no real intersections.
How to Use This Calculator
Enter the coefficients for the quadratic equation first. Use a, b, and c from y = ax² + bx + c.
Next, enter the line coefficients. Use L, M, and N from Lx + My = N.
Choose the decimal precision. Select whether complex intersections should appear. Press Calculate to view the result above the form.
Use the CSV button for spreadsheet output. Use the PDF button for a printable report.
Understanding Linear Quadratic Systems
What the System Means
A linear quadratic system compares one straight line with one parabola. The shared points are the solutions. Each solution has an x-value and a y-value. The line and parabola can meet twice, once, or never on the real plane. This calculator handles each case with direct algebra.
Why Substitution Works
The line gives a second expression for y. That expression can replace y in the quadratic equation. After replacement, only x remains. The new equation is quadratic in x. Solving it gives possible x-values. Each x-value is then placed into the line equation. That gives matching y-values.
Reading the Discriminant
The discriminant controls the shape of the answer. A positive value means two crossings. A zero value means one touch point. This is called tangency. A negative value means no real crossing. Complex mode still displays imaginary roots. These roots help advanced algebra study.
Using General Line Form
The calculator uses Lx + My = N. This form is flexible. It supports normal lines and vertical lines. Vertical lines do not have a regular slope. The calculator then solves x directly. After that, it evaluates the parabola at that x-value.
Advanced Output
The result includes more than points. It shows classification, the substituted equation, and the discriminant. When two real points exist, it also shows distance and midpoint. It can show tangent slopes as well. These details help with graphing, checking homework, and explaining solution behavior.
Practical Value
Linear quadratic systems appear in algebra, coordinate geometry, physics, economics, and modeling tasks. A line can represent a steady rate. A parabola can represent curved growth or projectile motion. Their intersections mark equal values. This makes the method useful for comparison problems and real applications.
Checking Accuracy
Always review the entered signs. A wrong negative sign changes the graph. Use enough decimal precision for final reports. For exact work, compare the displayed values with symbolic steps. The downloadable files help store results for later review and class notes.
FAQs
What is a linear quadratic system?
It is a system containing one line and one quadratic curve. The solutions are points where both equations have the same x and y values.
How many solutions can this system have?
It can have two real solutions, one real tangent solution, or no real solutions. Complex mode can show imaginary intersections when no real crossing exists.
What does the discriminant show?
The discriminant shows the solution type. Positive means two real intersections. Zero means one tangent point. Negative means no real intersection.
Can this calculator handle vertical lines?
Yes. Use M = 0 in Lx + My = N. The calculator solves x directly, then finds y from the parabola.
Why do I need coefficients L, M, and N?
They define the line in general form. This form works for sloped lines, horizontal lines, and vertical lines.
What happens when complex mode is off?
The calculator reports no real intersections when the discriminant is negative. It does not list imaginary coordinates.
Can I download my solution?
Yes. Use the CSV option for spreadsheet work. Use the PDF option for a simple printable solution report.
Is the midpoint always shown?
The midpoint is shown when two real intersections exist. It is not useful for one tangent point or complex-only results.