Question

Find the zeroes of the quadratic polynomial and verify the relationship between the zeroes and the coefficients.
${x^2} - 2x - 8$

Answer 1

Hint: Solve quadratic equations by factoring method to get the values of zeros. Now verify the relationship between the zeroes and the coefficients by finding the sum of its zeroes and product of its zeroes respectively.

Complete step-by-step answer:

${x^2} - 2x - 8$
$ \Rightarrow {x^2} - 4x + 2x - 8$
$ \Rightarrow x\left( {x - 4} \right) + 2\left( {x - 4} \right)$
$ \Rightarrow \left( {x - 4} \right)\left( {x + 2} \right)$
Zeroes are 4 and -2 .
Verification-

We know that the sum of zeroes are $\dfrac{{ - b}}{a}$ in the quadratic equation $a{x^2} + bx + c$
$ \Rightarrow 4 + \left( { - 2} \right) = - \dfrac{{ - 2}}{1}$
$ \Rightarrow 4 - 2 = 2$
$ \Rightarrow 2 = 2$
LHS = RHS

Product of zeroes = $\dfrac{c}{a}$
$ \Rightarrow \left( 4 \right)\left( { - 2} \right) = \dfrac{{ - 8}}{1}$
$ \Rightarrow - 8 = - 8$
LHS = RHS

Note: To solve such quadratic equations we need to recall how to do factorizing of a quadratic equation, also we should know how to express the quadratic equation in terms of sum of the squares and product of its squares.