Question

Verify the relation between zeroes and coefficients of the quadratic polynomial ${x^2} - 4$.

Answer 1

Hint: For zeroes factorize and for coefficients compare with the standard form of quadratic equation and use sum and product of the roots formula to verify.

In the problem we are given a quadratic polynomial ${x^2} - 4$.
We have to find the relation between zeroes and coefficients of this expression.
Zeroes of the quadratic polynomial in variable $x$are the values of $x$for which the given
expression value equals zero. These values are also called the roots of that quadratic equation.
Given the expression in the problem, equating it to $0$gives the quadratic equation as
${x^2} - 4 = 0{\text{ (1)}}$
A general quadratic equation is of the form
$a{x^2} + bx + c = 0{\text{ (2)}}$
Where $a$ is the coefficient of ${x^2}$, $b$ is the coefficient of $x$and $c$ is the constant.
From the properties of quadratic equation in the form of equation $(2)$,it is known that,
Sum of roots $ = \dfrac{{ - 2b}}{a}{\text{ (3)}}$
Products of roots $ = \dfrac{c}{a}{\text{ (4)}}$
First, we need to find the roots of the quadratic equation $(1)$.
Factorising equation $(1)$ using identity ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$, we
get,
$
{x^2} - {2^2} = 0 \\
\Rightarrow \left( {x - 2} \right)\left( {x + 2} \right) = 0 \\
\Rightarrow x = 2, - 2 \\
$
$ \Rightarrow x = 2$and$x = - 2$are the roots or zeros of the equation $(1)$

Now we need to convert equation $(1)$ into equation $(2)$form,
$
\Rightarrow \left( 1 \right){x^2} + \left( 0 \right)x - 4 = 0 \\
\Rightarrow a = 1,b = 0,c = - 4 \\
$
Using equations $(3)$and $(4)$ in above, we get,
Sum of roots $ = \dfrac{{ - 2b}}{a} = \dfrac{{ - 2 \times 0}}{1} = 0$
Products of roots $ = \dfrac{{ - 4}}{1} = - 4$
Also, since $x = 2$and$x = - 2$are the roots or zeros of the equation $(1)$,we get
Sum of roots $ = - 2 + 2 = 0$
Products of roots $ = - 2 \times 2 = - 4$
Since the results obtained above are in consensus with the previous results,
Hence relation $(3)$and $(4)$are verified, that is, relation between zeroes and coefficients of the
quadratic polynomial ${x^2} - 4$ are verified.

Note: Always try to remember the formula for sum of roots and product of roots for quadratic polynomials. Also, the polynomial should always be converted into general form as explained above before using the formulas.