Roots from Equation Calculator
Build quadratic equations from given roots
Building Equations from Roots
When you know the roots of a quadratic equation, you can construct the equation by working backwards from the factored form. This process is the reverse of factoring and uses the fundamental theorem that if r is a root, then (x - r) is a factor.
The Process
- Start with roots r₁ and r₂
- Write factors: (x - r₁)(x - r₂)
- Expand using FOIL or distribution
- Multiply by leading coefficient if needed
- Simplify to get standard form ax² + bx + c = 0
Connection to Vieta's Formulas
When expanding (x - r₁)(x - r₂), you get x² - (r₁ + r₂)x + r₁r₂. This shows that:
- The coefficient of x is the negative sum of roots
- The constant term is the product of roots
- These align perfectly with Vieta's formulas
Working with Complex Roots
This method works with complex roots too. If the roots are complex conjugates (a + bi and a - bi), the resulting equation will have real coefficients. The imaginary parts cancel out during multiplication, leaving a real quadratic equation.
The Leading Coefficient
The leading coefficient 'a' scales the entire equation but doesn't change the roots. Different values of 'a' give equivalent equations with the same roots but different coefficients for b and c. When a = 1, the equation is said to be monic.
Frequently Asked Questions
Why do we use (x - r) instead of (x + r)?
If r is a root, then when x = r, the expression must equal zero. For (x - r), when x = r, we get r - r = 0, which is correct. Using (x + r) would only give zero when x = -r.
Can I build an equation with any two numbers as roots?
Yes! Any two numbers (real or complex) can be roots of a quadratic equation. The resulting equation will always be valid, though it may have irrational or complex coefficients.
What if I have a repeated root?
If both roots are the same (r₁ = r₂ = r), the equation becomes a(x - r)² = 0, which expands to ax² - 2arx + ar² = 0. This is a perfect square trinomial.
Does the order of roots matter?
No, multiplication is commutative, so (x - r₁)(x - r₂) = (x - r₂)(x - r₁). You'll get the same equation regardless of which root you list first.
Why might I need to build an equation from roots?
This is useful in many scenarios: creating practice problems, modeling situations where you know desired outcomes, reverse-engineering equations, or verifying that certain values are indeed roots of a particular equation.
What happens if I use a = 0?
If a = 0, you no longer have a quadratic equation—it becomes linear or degenerate. The leading coefficient must be non-zero for a true quadratic equation.
Can I work backwards to verify my factoring?
Absolutely! If you factor an equation and get roots, you can use this method to expand back to standard form. If you get your original equation, your factoring was correct.
What if my roots are fractions or decimals?
The method works the same way. You'll get an equation with those roots, though the coefficients might be fractions or decimals. You can multiply through by a common denominator to get integer coefficients if desired.
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