Vieta's Formulas
Relates the roots of a quadratic to its coefficients.
Formula
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Example
For x² − 5x + 6 = 0: roots sum to 5 and multiply to 6 — they are 2 and 3.
How it works
Vieta’s formulas connect the coefficients of a quadratic ax² + bx + c = 0 to its roots x₁ and x₂ without solving the equation directly. They come from factoring the quadratic as a(x − x₁)(x − x₂) and then matching coefficients after expanding, which forces the root sum and product to take specific values. This is useful because it lets you reason about roots from the equation’s numbers, or build a quadratic when the roots are known.
When to use it
Use this when you need the roots’ sum or product, or when you know two numbers and want to form a quadratic equation with those numbers as roots. It is also helpful for checking answers to quadratic equations and for problems involving relationships between roots and coefficients.
Worked example, step by step
Find the roots of 3x² + 11x + 6 = 0 using Vieta’s formulas. Step 1: Identify the coefficients: a = 3, b = 11, c = 6. Step 2: Compute the sum of the roots: x₁ + x₂ = -b/a = -11/3. Step 3: Compute the product of the roots: x₁x₂ = c/a = 6/3 = 2. Step 4: Find two numbers that add to -11/3 and multiply to 2. Step 5: The numbers are -2/3 and -3, since (-2/3) + (-3) = -11/3 and (-2/3)(-3) = 2. Answer: the roots are x₁ = -2/3 and x₂ = -3.
Common mistakes to avoid
- Forgetting the minus sign in the sum formula, so using b/a instead of -b/a
- Using c instead of c/a when the quadratic is not monic
- Assuming Vieta’s formulas give the roots directly without checking that the numbers satisfy both sum and product
Did You Know?
Vieta's formulas link a polynomial's roots to its coefficients without ever solving the equation.
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