Calculator
Formula Used
The calculator starts with the standard quadratic equation:
ax² + bx + c = 0
The main formula is:
x = (-b ± √(b² - 4ac)) / 2a
The discriminant is:
D = b² - 4ac
If D is positive, two real roots exist. If D is zero, one repeated real root exists. If D is negative, two complex roots exist.
The vertex is found with x = -b / 2a. Then substitute that value into the equation to find the matching y-value.
How To Use This Calculator
- Enter the value of coefficient a.
- Enter the value of coefficient b.
- Enter the value of coefficient c.
- Choose the number of decimal places.
- Change the variable symbol if needed.
- Press Calculate to view roots and work steps.
- Use Download CSV or Download PDF to save the result.
Example Data Table
| a | b | c | Equation | Expected result |
|---|---|---|---|---|
| 1 | -5 | 6 | x² - 5x + 6 = 0 | x = 3, x = 2 |
| 1 | 2 | 1 | x² + 2x + 1 = 0 | x = -1 repeated |
| 1 | 2 | 5 | x² + 2x + 5 = 0 | x = -1 ± 2i |
| 2 | -4 | -6 | 2x² - 4x - 6 = 0 | x = 3, x = -1 |
Understanding Quadratic Equations
A quadratic equation has the form ax² + bx + c = 0. The value of a cannot be zero. This shape creates a parabola when graphed. The roots show where that curve crosses the horizontal axis. These roots may be real, repeated, or complex.
Why Work Steps Matter
Many calculators show only the final roots. That is useful, but it hides the reasoning. A worked solution shows each substitution, square, product, and simplification. This helps students check signs and coefficients. It also makes mistakes easier to find before homework is submitted.
Role of the Discriminant
The discriminant is b² - 4ac. It controls the type of answer. A positive value gives two real roots. A zero value gives one repeated root. A negative value gives two complex roots. This calculator displays that decision before showing the final values, so the solution path stays clear.
Graph Details
The same coefficients also describe the graph. The vertex gives the turning point of the parabola. The axis of symmetry runs through that point. The y-intercept equals c. These details help connect algebra with graphing. They are useful when checking a plotted answer or sketching a curve quickly.
Better Study Records
Export options make the tool practical for learning. The CSV file stores the main numbers in a simple table. The PDF file saves the equation, roots, and work notes in a clean report. These files can support study logs, class notes, or tutoring records.
Practical Accuracy Tips
Use enough decimal places for your task. Rounding too early can change a final answer. Keep exact radical forms when possible. Compare the sum and product of roots with -b/a and c/a. This gives a fast accuracy check and reinforces the relationship between coefficients and solutions.
Where This Tool Helps
The calculator is helpful for classwork, test review, and quick checking. It can support factoring lessons, graphing lessons, and word problems. Users can enter any real coefficients and inspect the complete path. The displayed work encourages understanding, not guessing. It also shows related values that teachers often request, such as the vertex, axis, intercept, root sum, and root product. This makes one page useful for both algebra practice and graph analysis daily sessions.
FAQs
What is a quadratic equation?
A quadratic equation is an equation with the highest power of the variable equal to two. Its standard form is ax² + bx + c = 0, where a cannot be zero.
What does the discriminant show?
The discriminant shows the root type. A positive value gives two real roots. Zero gives one repeated real root. A negative value gives two complex roots.
Can this calculator show complex roots?
Yes. When the discriminant is negative, the calculator shows roots using i. It also explains why the roots are complex in the worked steps.
Why must a not equal zero?
If a equals zero, the equation no longer has a squared term. It becomes linear, not quadratic. The calculator handles that case separately.
What is the vertex of a quadratic?
The vertex is the turning point of the parabola. It is found using x = -b / 2a, then substituting that value into the equation.
What is the axis of symmetry?
The axis of symmetry is the vertical line through the vertex. For a quadratic equation, it is x = -b / 2a.
Can I export my result?
Yes. After calculation, use the CSV button for spreadsheet data. Use the PDF button for a clean report with values and working notes.
Does rounding affect the answer?
Rounding can slightly change displayed roots, especially with long decimals. Increase decimal places when you need more accurate study or checking results.