Calculator
Use the form y = ax² + bx + c and y = mx + n.
Example Data Table
| Quadratic | Line | Discriminant | Expected result |
|---|---|---|---|
| y = x² - 4x + 3 | y = x - 1 | 9 | Two intersections |
| y = x² | y = 0 | 0 | One tangent point |
| y = x² + 1 | y = 0 | -4 | No real crossing |
| y = 2x + 1 | y = -x + 4 | Not applicable | One line intersection |
Formula Used
The calculator starts with these equations:
Quadratic: y = ax² + bx + c
Linear: y = mx + n
Set both right sides equal:
ax² + bx + c = mx + n
Move all terms to one side:
Ax² + Bx + C = 0
Where A = a, B = b - m, and C = c - n.
The discriminant is:
D = B² - 4AC
The root formula is:
x = (-B ± √D) / 2A
Then y is found with:
y = mx + n
How to Use This Calculator
Enter the coefficients for the quadratic equation.
Enter the slope and intercept for the linear equation.
Set the decimal precision and tolerance values.
Add a domain range when your problem has limits.
Select complex results when you want non-real answers.
Press Calculate to view intersections above the form.
Use CSV for spreadsheet records.
Use PDF for a printable report.
Quadratic Linear System Guide
Basic Meaning
A quadratic linear system compares a parabola with a straight line. The solution is every point where both equations are true. In this calculator, the quadratic equation is written as y = ax² + bx + c. The linear equation is written as y = mx + n. When the two expressions are set equal, one quadratic equation in x is created.
Why Intersections Matter
Intersections show shared values. They can describe projectile paths, demand curves, design limits, break even points, and classroom algebra problems. A system may have two real intersections, one tangent point, or no real intersection. The discriminant explains the count before coordinates are listed. A positive value gives two points. Zero gives one repeated point. A negative value gives complex x values, which means no real crossing on the coordinate plane.
Method Used
The calculator moves the line to the quadratic side. It solves ax² + bx + c = mx + n. This becomes ax² + (b - m)x + (c - n) = 0. The roots are found with the quadratic formula. Each x value is then placed into the linear equation to find y. A check value is also shown by substituting the same x into both equations.
Advanced Options
Decimal precision controls how many digits appear in the answer. The tolerance field helps decide whether tiny computer round off differences are treated as zero. The domain fields let you mark a practical x range, so each point can be labeled inside or outside your working interval. The calculator also reports the vertex of the parabola and the line slope. These details help explain the shape and the final result.
Practical Use
Start with coefficients from your problem. Use negative signs where needed. Press calculate to view the result box above the form. Review the discriminant and points first. Then compare the substitution checks. Download a CSV file when you need spreadsheet data. Use the PDF button for a printable summary. The example table gives sample cases for two intersections, tangent contact, and no real crossing.
It is useful to keep exact coefficients during entry. Rounded inputs can slightly move the final points. For homework, copy the formula line and the check values into your written solution before submitting final answers carefully.
FAQs
What does this calculator solve?
It solves systems containing one quadratic equation and one linear equation. It finds their intersection points, checks each coordinate, and reports the discriminant.
How many answers can a system have?
It can have two real intersections, one repeated intersection, no real intersection, or infinite solutions when both equations represent the same line.
What is the discriminant?
The discriminant is D = B² - 4AC. It tells whether the reduced equation has two, one, or no real roots.
Why is y calculated from the line?
After x is found, y = mx + n is direct and simple. The calculator also checks the same x in the quadratic equation.
What does tolerance mean?
Tolerance decides when a very small number should count as zero. It helps avoid misleading results from decimal rounding.
Can it show complex solutions?
Yes. Select the complex result option. Complex points appear when the line does not cross the parabola in real coordinates.
Why use a domain range?
A domain range helps mark whether a solution is inside your practical interval. This is useful for applied problems with limits.
What can I download?
You can download a CSV file for spreadsheet use. You can also download a simple PDF report for printing or sharing.