Theory of Equations — Vieta's Formulae explained through a friendly whiteboard conversation, examples, visuals, and practice for school students.
Step 1
Notes
Step 2
Formulae
Step 3
Examples
Step 4
Practice
Revise the definition, formulae, important results, and exam method first.
Use the sticky diagram while this dedicated tool is being prepared.
Solve only this concept and let the app adapt from your answers.
You have met the idea. Now open a tiny puzzle, a useful fact, and the next best action.
Continue the journey
Vieta's formulae connect the roots of an equation with its coefficients.
They help solve questions without finding each root first.
They are used in quadratic equations, polynomial equations, and exatyle algebra.
For a quadratic equation a, the roots are often called and .
The sum of roots i.
The product of roots is .
If the roots are known, the equation can be written as roduct of root for a monic quadratic.
These relationships come from comparing coefficients.
Meaning: Roots are values that make an equation equal to zero.
Definition: If , then r is a root of f(x).
For , roots are 2 and 3.
Meaning: Add the two roots.
Definition: For a, + = -.
The negative sign before b is important.
Meaning: Multiply the two roots.
Definition: For a, = c/a.
Use the constant term divided by leading coefficient.
Meaning: Build a quadratic from root information.
Definition: , where S is sum and P is product.
If roots are 4 and 5, and , so .
Easy
Question: Find sum and product of roots of .
Medium
Question: Form a quadratic equation with roots 3 an.
Exam-level
Question: If roots of have su, find k.
Forgetting the minus sign in sum
Why it happens: - is remembered as .
Correct approach: Write the formula before substituting.
Using c instead of
Why it happens: Leading coefficient is ignored.
Correct approach: Divide by a when a is not 1.
Putting plus Sx in formed equation
Why it happens: The standard form is not checked.
Correct approach: Use .
Do I need to solve the quadratic to use Vieta?
No. Vieta uses coefficients directly.
What if a is not 1?
Us and .
Are roots and zeroes the same?
Yes, in this context they mean values that make the polynomial zero.
Practise the same concept from this article. Your student profile may guide entitlement and follop recommendations, but it does not replace this concept.