Calculator
Example Data Table
| a | b | c | Line | Discriminant | Expected result |
|---|---|---|---|---|---|
| 1 | -1 | -6 | y = x | 28 | Two real intersections |
| 1 | 0 | 0 | y = 0 | 0 | One tangent intersection |
| 1 | 0 | 3 | y = 0 | -12 | No real intersection |
Formula Used
The calculator starts with a parabola and a line.
Quadratic: y = ax² + bx + c
Line: y = mx + k
Set both y values equal: ax² + bx + c = mx + k.
Move all terms to one side: ax² + (b - m)x + (c - k) = 0.
Then use D = B² - 4AC, where A = a, B = b - m, and C = c - k.
If D is positive, two real intersections exist. If D is zero, the line is tangent. If D is negative, no real intersection exists.
How to Use This Calculator
Enter the parabola coefficients a, b, and c. Choose the line form. Use slope intercept form for y = mx + k. Use standard form for Lx + My + N = 0. Pick decimal places and tolerance. Press calculate. Read the result section above the form. Download the CSV or PDF when you need a saved copy.
Algebraic System Guide
Why This Calculator Helps
A linear quadratic system joins one straight line with one parabola. The solution is any ordered pair that satisfies both equations. This calculator solves that pair algebraically. It avoids guessing from a graph. It also shows the substituted equation, discriminant, vertex, and final points. These details make checking easier.
Understanding the Method
The main step is substitution. The line gives a value for y. The parabola also gives a value for y. When both expressions are equal, their x values must match. Moving every term to one side creates a quadratic equation in x. The discriminant then predicts the number of real solutions before the roots are listed.
Reading the Discriminant
A positive discriminant means the line cuts the parabola twice. A zero discriminant means the line touches the curve once. A negative discriminant means the line misses the parabola in real coordinates. Complex roots can still appear in the algebra. They are useful for advanced study, but they are not visible intersections.
Standard Line Support
Many textbooks give a line in standard form. This tool converts that form when possible. If the y coefficient is not zero, it rewrites the equation as slope intercept form. If the line is vertical, substitution uses a fixed x value instead. That gives one point on a function parabola.
Practical Uses
Students can use the page to test homework. Teachers can create examples with exact steps. Designers can compare a straight path with a curved model. The export buttons help save work for notes, reports, or review sheets. Use sensible precision when decimals are long. Use tighter tolerance when near tangent results matter.
Accuracy Notes
Small rounding changes may affect a near zero discriminant. The tolerance field controls that decision. A larger tolerance treats very small values as zero. A smaller tolerance separates close intersections more strictly. Always compare the displayed substitution equation with the original system. This habit catches typing mistakes early.
Checking Your Answers
After solving, substitute each listed point into both original equations. The y values should match after rounding. When they do not match, check signs, coefficients, and the chosen line form. Standard form errors often come from moving N to the other side. Save the result table after checking, so later revisions have a reliable reference during homework review or classroom lesson planning sessions.
FAQs
What is a linear quadratic system?
It is a system containing one linear equation and one quadratic equation. Its solutions are points where the line and parabola share 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 solution. A vertical line with a function parabola gives one real point.
Why does the discriminant matter?
The discriminant tells the real solution count. Positive means two intersections. Zero means one tangent point. Negative means no real intersection.
Can I enter a standard form line?
Yes. Select standard form and enter L, M, and N for Lx + My + N = 0. The calculator converts it when possible.
What happens with a vertical line?
A vertical line has a fixed x value. The calculator substitutes that x into the quadratic equation and returns the matching y value.
Why do I see complex roots?
Complex roots appear when no real intersection exists. They solve the transformed quadratic algebraically, but they are not graph points on the real plane.
What tolerance should I use?
Use a small tolerance for normal problems. Increase it only when rounding makes a tangent case appear slightly positive or negative.
Can I save the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable summary of equations, steps, and solutions.