Question

How do you write a quadratic function with zeros $-8$ and $-2$?

Answer 1

Hint: In this question we have been given two numbers which represent the zeroes of a quadratic function which we need to create. We will consider the quadratic function as $f\left( x \right)$. We know that a quadratic equation is an equation with degree $2$ which has $2$ roots. We will consider the roots to be $-8$ and $-2$ , write them in the form of a factor of the quadratic equation, and then we will multiply both the factors to get the required quadratic equation.

Complete step by step answer:
Consider the function to be $f\left( x \right)$.
We have the zeroes of $f\left( x \right)$ given as $-8$ and $-2$, this implies that $f\left( -8 \right)=0$ and $f\left( -2 \right)=0$.
Therefore, we can write the roots of the equation as:
$\Rightarrow x=-8$ and $x=-2$
On transferring the terms from the right-hand side to the left-hand side, we get:
$\Rightarrow x+8=0$ and $x+2=0$
Now the quadratic equation can be derived from the solutions of it by multiplying them therefore, the quadratic equation is:
$\Rightarrow \left( x+8 \right)\left( x+2 \right)$
Now on multiplying the terms, we get:
$\Rightarrow {{x}^{2}}+2x+8x+16$
On simplifying the terms, we get:
$\Rightarrow {{x}^{2}}+10x+16$, which is the required quadratic equation with zeros $-8$ and $-2$.

Note: It is to be remembered that zeros of the equation represent the root or the solution of the equation, it is the term which when substituted in the equation, we get the value as $0$.
The roots of a quadratic equation can be found using the formula $(x,y)=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2ac}$
Where $(x,y)$ are the roots of the equation and $a,b,c$ are the coefficients of the terms in the quadratic equation.