How to Build Quadratics from Roots?

A step-by-step guide to building quadratics from roots

If \(α\) and \(β\) are the two roots of a quadratic equation, the formula for constructing the quadratic equation is:

\(\color{blue}{x^2-(α + β)x + αβ = 0}\)

That is,

\(\color{blue}{x^2\:-\:\left(sum\:of\:roots\right)x\:+\:product\:of\:roots\:=\:0}\)

If a quadratic equation is given in the standard form, we can find the sum and product of the roots using the coefficients of \(x^2, x\), and the constant term.

Let’s consider the standard form of a quadratic equation,

\(ax^2+bx+c=0\)

Where \(a, b\) and \(c\) are real and rational numbers.

Let \(α\) and \(β\) be the two zeros of the quadratic equation above. Then the formula for obtaining the sum and the product of the roots of a quadratic equation is:

\(\color{blue}{α+β=-\frac{b}{a}=-\frac{coefficient\:of\:x}{coefficient\:of\:x^2\:}}\)

\(\color{blue}{αβ=\frac{c}{a}=\frac{constant\:term}{coefficient\:of\:x^2\:}}\)

Note: The irrational roots of a quadratic equation occur as conjugate pairs. That is if \((m\:+\sqrt{n})\) is a root, then \((m\:-\sqrt{n})\) is the other root of the same quadratic equation.

Build Quadratics from Roots – Example 1:

Build the quadratic equation whose roots are \(2\) and \(-\frac{1}{2}\).

Solution:

The sum of the roots is: \(2+ (-\frac{1}{2})= \frac {3}{2}\)

The product of the roots is: \( 2\times (-\frac{1}{2})= -1\)

Formation of the quadratic equation: \(x^2\:-\:\left(sum\:of\:roots\right)x\:+\:product\:of\:roots\:=\:0\)

So, the quadratic equation is: \(x^2- \frac{3}{2}x -1=0\)