Calculator input
Formula used
If the roots are r₁ and r₂, and the leading coefficient is a, the factored equation is:
a(x - r₁)(x - r₂) = 0
After expansion, the standard quadratic form is:
ax² - a(r₁ + r₂)x + a(r₁r₂) = 0
So the coefficients are A = a, B = -a(r₁ + r₂), and C = a(r₁r₂).
How to use this calculator
- Enter the leading coefficient. It cannot be zero.
- Enter both roots. Use imaginary parts for complex roots.
- Choose the variable symbol and decimal precision.
- Select the output style you need.
- Press the calculate button to see the result above the form.
- Download the result as CSV or PDF when needed.
Example data table
| Leading coefficient | Root 1 | Root 2 | Expanded equation |
|---|---|---|---|
| 1 | 2 | 5 | x² - 7x + 10 = 0 |
| 3 | -1 | 4 | 3x² - 9x - 12 = 0 |
| 2 | 4 + 3i | 4 - 3i | 2x² - 16x + 50 = 0 |
Understanding Roots and Coefficients
A quadratic equation is often written as ax² + bx + c = 0. The roots are the x values that make the equation equal zero. When the roots are known, the equation can be rebuilt quickly. You only need both roots and the leading coefficient. The leading coefficient controls the vertical stretch and direction. Positive values open the parabola upward. Negative values open it downward.
Why This Conversion Matters
Many algebra tasks move between roots and coefficients. A graph may show two intercepts. A problem may give two solutions. A design model may require an equation from target values. This calculator connects those forms. It uses the factor form first. Then it expands the expression into standard form. This helps users check work, prepare answers, and compare related equations.
Role of the Leading Coefficient
The same roots can produce many quadratic equations. For example, roots 2 and 5 give x² - 7x + 10 = 0 when a equals 1. If a equals 3, the equation becomes 3x² - 21x + 30 = 0. The roots stay the same. The graph becomes steeper. The coefficient changes the scale, not the solution locations.
Vieta Relationships
Vieta rules give a fast coefficient check. For ax² + bx + c = 0, the sum of roots equals -b/a. The product of roots equals c/a. These relationships explain every expanded result. They also reveal mistakes. If the shown sum or product does not match your expected roots, review the signs and values entered.
Real and Complex Roots
Real roots create real coefficients when the leading coefficient is real. Complex roots can also create real coefficients when they are conjugates. A pair such as 4 + 3i and 4 - 3i works cleanly. If the complex roots are not conjugates, the calculator still builds the equation. It warns that coefficients may be complex. This is useful in advanced algebra and engineering work.
Interpreting the Graph
Every result also describes a parabola. The roots are x-intercepts when they are real. The coefficient a sets the opening direction. The vertex gives the highest or lowest point. The axis of symmetry passes through the vertex. These details make the equation easier to sketch. They also help when comparing models, trajectories, and revenue curves.
Checking Your Answer
Substitute each root into the final equation. The value should become zero. Small decimal differences can appear after rounding. Use more decimal places when inputs include long decimals, real data, or measured values.
Using Results Wisely
The expanded equation is best for standard algebra steps. The factored equation is best for seeing the roots. The discriminant helps describe the root type. The vertex shows the turning point when coefficients are real. Always choose enough decimal places for your task. For exact classroom work, enter integers or simple decimals. For measurements, use the precision supported by your data.
FAQs
What does this calculator create?
It creates a quadratic equation from two roots and a leading coefficient. It shows factored form, expanded form, coefficients, Vieta checks, discriminant, and vertex details when available.
What is the required leading coefficient?
The leading coefficient can be any real number except zero. A zero value would remove the squared term, so the result would no longer be a quadratic equation.
Can I use decimal roots?
Yes. Enter decimal roots directly in the real part fields. Then choose enough decimal places to keep the displayed coefficients useful for your work.
Can it handle complex roots?
Yes. Use the imaginary part fields for complex roots. Conjugate complex roots usually create real coefficients when the leading coefficient is real.
Why are my coefficients complex?
Complex coefficients can appear when entered complex roots are not conjugates. For real coefficients, use roots like 3 + 2i and 3 - 2i.
What is Vieta's relationship?
For ax² + bx + c = 0, the sum of roots equals -b/a. The product of roots equals c/a. The calculator displays both checks.
What is the factored form?
The factored form is a(x - r₁)(x - r₂) = 0. It clearly shows the roots used to build the quadratic equation.
What is the expanded form?
The expanded form is ax² + bx + c = 0. It is the standard form often required for algebra, graphing, and comparison tasks.
Why does the vertex sometimes not show?
The vertex is shown for real coefficients. If coefficients are complex, the usual real coordinate vertex is not available in the same graphing sense.
Can I evaluate the equation at a point?
Yes. Enter a value in the evaluate field. The calculator substitutes that value into the final equation and shows the resulting value.
Do different leading coefficients change the roots?
No. Different nonzero leading coefficients create related quadratic equations. They change graph stretch and direction. The roots remain unchanged in every properly scaled version.