In this article, Mathema, an educational platform, explores Vieta’s Theorem and its applications in factored quadratic equations. Designed for 8th-grade students and their parents, this article aims to simplify complex mathematical concepts.
Vieta’s theorem for factored quadratic equations (a=1)
These are formulas that express the coefficients of a polynomial in terms of its roots. The theorem is named after François Viète, a French mathematician. Vieta’s theorem is usually used to solve factored quadratic equations, i.e., when the coefficient a=1. Here’s how Vieta’s theorem is formulated for a factored quadratic equation:
The sum of the roots of a factored quadratic equation is equal to the second coefficient with the opposite sign, and the product of the roots is equal to the constant term.
Vieta’s theorem formula for factored quadratic equations
Here’s what the Vieta’s theorem formula looks like for a factored quadratic equation:
\[x^2+px+q=0,\;x_1\cdot x_2=q;\;x_1+x_2=-p\]This formula of Vieta’s theorem allows one to compute their sum and product without knowing the roots of the quadratic trinomial.
Application of Vieta’s theorem
Let’s analyze Vieta’s theorem and the formula of Vieta’s theorem with a simple example. This will help understand how to correctly solve factored quadratic equations using them.
How to find the roots of an equation using Vieta’s theorem?
\[x^2-5x+6=0\]Solution: According to Vieta’s theorem, we write:
\[x_1+x_2=5\] \[x_1\cdot x_2=6\]We find the values of 𝑥 that satisfy the equation:
\[x_1=\;2;\;x_2=3\]Answer: the roots of the equation are 2 and 3.
Inverse Vieta’s theorem
The concept of the inverse Vieta’s theorem speaks for itself, as it works exactly the opposite way. Using the inverse theorem to Vieta’s theorem, you can find the roots of factored quadratic equations.
If two numbers are such that their sum equals the second coefficient of the factored quadratic equation taken with the opposite sign, and their product equals its constant term, then these numbers are the roots of this factored quadratic equation.
How to apply the inverse Vieta’s theorem
Task: Formulate a quadratic equation, knowing its roots.
\[x_1=\;3;\;x_2=-1\]Solution: Let the sought quadratic equation be:
\[x^2+px+q=0\]Using the inverse Vieta’s theorem, we get the following relationships:
\[x_1\cdot x_2=q;\;x_1+x_2=-p\]Now let’s find the values of 𝑝 and 𝑞:
\[p=-(x_1+x_2)=-(3+(-1))=-2\] \[q=x_1\cdot x_2\;=\;3\cdot(-1)=-3\]Answer: the equation we were looking for looks like this:
\[x^2-2x-3=0\]
