Write A Quadratic Equation with Given Roots Calculator
'/%3E%3Ccircle cx='48' cy='39' r='19' fill='%23f2c9a5'/%3E%3Cpath d='M27 35c3-16 39-17 42 2-8-8-27-9-42-2z' fill='%23374151'/%3E%3Cpath d='M21 86c5-24 49-28 56 0z' fill='%232563eb'/%3E%3Ccircle cx='41' cy='41' r='2' fill='%23111827'/%3E%3Ccircle cx='55' cy='41' r='2' fill='%23111827'/%3E%3Cpath d='M42 52c4 3 9 3 13 0' stroke='%2392400e' stroke-width='3' fill='none' stroke-linecap='round'/%3E%3C/svg%3E)
Reviewed by Calculator Editorial Team
This calculator helps you quickly write a quadratic equation when you know its roots. Whether you're a student studying algebra or a professional working with quadratic functions, this tool provides an efficient way to generate the equation from given roots.
Introduction
A quadratic equation is a second-degree polynomial equation in a single variable. It has the general form:
General Form of Quadratic Equation
ax² + bx + c = 0
Where a, b, and c are coefficients, and x is the variable. The roots of the equation are the values of x that satisfy the equation.
If you know the roots of a quadratic equation, you can write the equation in its factored form:
Factored Form of Quadratic Equation
a(x - r₁)(x - r₂) = 0
Where r₁ and r₂ are the roots of the equation. This calculator uses this factored form to generate the quadratic equation from given roots.
How to Use the Calculator
Using the calculator is simple:
- Enter the first root in the "First Root" field.
- Enter the second root in the "Second Root" field.
- Optionally, enter a coefficient for the quadratic term (a) if it's not 1.
- Click the "Calculate" button to generate the quadratic equation.
- The result will be displayed in both factored and expanded forms.
You can also reset the calculator to start over by clicking the "Reset" button.
The Formula
The formula used to write a quadratic equation with given roots is based on the factored form of the equation:
Quadratic Equation from Roots
a(x - r₁)(x - r₂) = 0
Where:
- a is the coefficient of the quadratic term (default is 1 if not specified)
- r₁ is the first root
- r₂ is the second root
To convert this to the standard form, you expand the equation:
Expanded Form
ax² - a(r₁ + r₂)x + a r₁ r₂ = 0
Worked Example
Let's say you have roots at x = 2 and x = 3, and you want to write the quadratic equation with a coefficient of 2 for the quadratic term.
Using the factored form:
Factored Form
2(x - 2)(x - 3) = 0
Expanding this:
Expanded Form
2x² - 2(2 + 3)x + 2(2)(3) = 0
2x² - 10x + 12 = 0
So the quadratic equation is 2x² - 10x + 12 = 0.
Frequently Asked Questions
What is a quadratic equation?
A quadratic equation is a second-degree polynomial equation in a single variable. It has the general form ax² + bx + c = 0, where a, b, and c are coefficients, and x is the variable.
How do I find the roots of a quadratic equation?
The roots of a quadratic equation can be found using the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a). The roots are the values of x that satisfy the equation.
Can I use this calculator for complex roots?
Yes, this calculator can handle complex roots. When you enter complex roots, the calculator will generate the corresponding quadratic equation.
What if I only have one root?
If you only have one root, the quadratic equation will have a double root at that value. The equation will be in the form a(x - r)² = 0, where r is the root.
How do I interpret the result?
The result shows the quadratic equation in both factored and expanded forms. The factored form shows the roots explicitly, while the expanded form is the standard quadratic equation.