# 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: $$x^2- \frac{3}{2}x -1=0$$

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