Vieta’s Formulas

Additional
Two simple and very useful formulas that connect the roots of a quadratic trinomial with its coefficients. They let you check roots quickly, build equations, and study special kinds of quadratic equations.

Connections:

Writing a quadratic trinomial as a product of factors leads straight to Vieta’s formulas, so you absolutely need to know how factoring works.
Factoring a quadratic trinomial
Factoring by hand

Statistics:

  • Statement1
  • Important3
  • Problem17
  • Updated
Article
Summary
Practice

Nice coefficients

Method
😀
Elementary
Construct a quadratic equation with integer coefficients if one of its roots is known to be:
7

Letter test

😀
Elementary
For which values of p and q are the roots of the equation x2+px+q=0x^2 + px + q = 0 equal to 2p and q2\dfrac{q}{2}?

Another derivation of Vieta’s formulas

😀
Elementary
Derive Vieta’s formulas from the quadratic formula for the roots of a quadratic equation:
x1,2=−B±D2Ax_{1,2} = \frac{-B \pm \sqrt{D}}{2A}

Balancing the roots

🤔
Intermediate
In the equation x2−4x+a=0x^2-4x+a=0, the sum of the squares of the roots is 16. Find a.

Twisted sum

🤔
Intermediate
Two distinct numbers a and b satisfy the following chain of equalities:
a2+3a+1=b2+3b+1=0a^2 + 3a + 1 = b^2 + 3b + 1 = 0
Find the value of the following expression:
ab+ba\frac{a}{b} + \frac{b}{a}

A root from a root

Method
🤔
Intermediate
Let x1x_1 and x2x_2 be the roots of a quadratic equation in general form Ax2+Bx+C=0Ax^2 + Bx + C = 0. Assume that neither the roots nor the coefficients are zero. Construct a new equation whose roots are:
  • 1
    Opposite to the roots of the original equation
  • 2
    Reciprocal to the roots of the original equation
  • 3
    5 times the roots of the original equation
    • Applying the formulas you get, write down 3 new equations obtained from 3x2−x−1=03x^2 - x - 1 = 0.

    Root transformer

    🤔
    Intermediate
    Let x1x_1 and x2x_2 be the roots of the quadratic equation 2x2−7x−3=02x^2 - 7x - 3 = 0. Construct a new quadratic equation whose roots are the numbers:
    x1+1x2x_1 + \dfrac{1}{x_2} and x2+1x1x_2 + \dfrac{1}{x_1}

    Acrobatic roots

    🤔
    Intermediate
    Without calculating the roots of the equation 3x2+8x−1=03x^2 + 8x - 1 = 0, find:
    x12+x22x_1^2 + x_2^2

    Roots and parameters

    🤔
    Intermediate
    It is known that the roots of the equation x2−5x+a=0x^2 - 5x + a = 0 are 1 less than the roots of the equation x2−7x+3a−6=0x^2 - 7x + 3a - 6 = 0. Find a and the roots of each equation.

    Checking the quadratic formula

    🤯
    Advanced
    Let the roots of a quadratic equation be denoted by n and m. Express the following expression, built from the coefficients of that same equation, in terms of those roots:
    B2−4ACA2\frac{B^2 - 4AC}{A^2}

    Root difference

    🤯
    Advanced
    Without calculating the roots of the equation 2x2−5x+1=02x^2 - 5x + 1 = 0, find just the difference and the difference of the squares of its roots. Why do both give two possible answers?

    A doubled root

    🤯
    Advanced
    For which values of the parameter a is one root of the quadratic equation twice the other?
    (a2−5a+3)x2+(3a−1)x+2=0(a^2 - 5a + 3)x^2 + (3a-1)x + 2 = 0

    I’m out of root-themed titles

    🤯
    Advanced
    The numbers n and m are the roots of the quadratic equation x2+5x+3=0x^2 + 5x + 3 = 0. Without calculating the values of those roots, determine what quadratic equation will have the following roots:
    (n−1n)2and(m−1m)2\left( n - \frac{1}{n} \right)^2 \quad \text{and} \quad \left( m - \frac{1}{m} \right)^2

    Zero-sum game

    🤯
    Advanced
    Using only Vieta’s formulas, figure out what the roots of a quadratic equation are if the sum of its coefficients is equal to zero.
    Vieta’s Formulas