Question

How can you u check whether 2/3and -1/2are the roots of 6x^2-x-2=0?​

Answer 1

Answer:

Yes, ⅔ and -½ are the roots of 6x² - x - 2 = 0.

Explanation:

To check whether ⅔ and -½ is zero or not we'll do verification of relationship between roots of the equation.

Given equation,

$\rm \implies \: {6x}^{2} - x - 2 = 0$

On comparing with the general form of a quadratic equation ax² + bx + c = 0

We get,

• a = 6
• b = -1
• c = -2

Given roots of the polynomial are :-

$\rm = \dfrac{2}{3} \: and \: \dfrac{ - 1}{2}$

We know that,

$\rm \bullet \: \: sum \: of \: roots = \dfrac{ - b}{a}$

Then,

$\rm \implies \: \dfrac{2}{3} + \dfrac{ - 1}{2} = \dfrac{ - ( - 1)}{6}$

$\rm \implies \: \dfrac{4 - 3}{6} = \dfrac{1}{6}$

$\rm \implies \: \dfrac{1}{6} = \dfrac{1}{6}$

And also,

$\rm \bullet \: \: product \: of \: roots = \dfrac{c}{a}$

So,

$\rm \implies \: \dfrac{2}{3} \ast \dfrac{ - 1}{2} = \dfrac{ - 2}{6}$

$\rm \implies \: \dfrac{ \not2}{3} \ast \dfrac{ - 1}{ \not2} = \dfrac{ - 2}{6}$

$\rm \implies \: \dfrac{ - 1}{3} = \dfrac{ - 1}{3}$

Hence, verified!

Answer 2

Given :

Quadratic equation is 6x² - x - 2 = 0

To find :

Whether ⅔ and -½ are the roots of the given equation

Solution :

Before solving this problem, we must have some basic knowledge about quadratic equations.

A quadratic equation is an equation in one variable where the highest power of the variable is 2.

General form of a quadratic equation is :

• ax² + bx + c = 0

Sum of zeroes = -b/a

Product of zeroes = c/a

Here,

• b = Coefficient of x
• a = Coefficient of x²
• c = Constant term

By comparing the given equation with general form of quadratic equation, we get :

• a = 6
• b = -1
• c = -2

There are basically three different ways to prove whether the given roots are the roots of given quadratic equation or not.

• By verifying product of roots
• By verifying sum of roots
• By substituting the roots in given equation

Let's discuss all the methods one by one!

Method-1

• By verifying product of roots

If the product of given roots be equal to c/a where a and c are coefficient of x² and constant term respectively, will imply that the given values are exactly the roots of given equation.

$\sf Product \: of \: roots=\dfrac{c}a=\dfrac{ - 1}{2}\times\dfrac{2}{3}$

$\sf Product \: of \: roots=\dfrac{ - 2}6=\dfrac{ - 1}{3}$

$\sf Product \: of \: roots=\dfrac{ - 1}3=\dfrac{ - 1}{3}$

Since product of zeroes are verified this means that gives roots are correct.

Method-2

• By verifying sum of zeroes

If the sum of given roots of equation be equal to -b/a where a and b are the coefficient of x² and x respectively, will imply that given roots are correct.

$\sf Sum \: of \: roots =- \dfrac{b}{a}=-\dfrac{1}2+\dfrac{2}3$

$\sf Sum \: of \: roots =- \dfrac{( - 1)}{6}= \dfrac{ - 3 + 4}{6}$

$\sf Sum \: of \: roots = \dfrac{1}{6}= \dfrac{ 1}{6}$

Hence sum of zeroes are verified this means that gives roots are correct.

Method-3

• By substituting the given roots in equation

This is the most common method to check whether given roots are actually the roots of equation or not. In this method, we will substitute the given roots in given equation and will check if the result we get is equal to 0 or not.

Roots are actually those values, which on substituting with x, results equal to 0.

For root 2/3

$\sf\: : \implies 6{x}^{2} - x - 2 = 0$

Substituting x = 2/3

$\sf\: : \implies 6{ \left( \dfrac{2}{3} \right)}^{2} - \dfrac{2}{3} - 2 = 0$

$\sf\: : \implies 6 \times \dfrac{4}{9} - \dfrac{2}{3} - 2 = 0$

$\sf\: : \implies \dfrac{2 \times 4}{3} - \dfrac{2}{3} - 2 = 0$

$\sf\: : \implies \dfrac{8}{3} - \dfrac{2}{3} - 2 = 0$

$\sf\: : \implies \dfrac{8 - 2}{3} - 2 = 0$

$\sf\: : \implies \dfrac{6}{3} - 2 = 0$

$\sf\: : \implies 2 - 2 = 0$

$\sf\: : \implies 0 = 0$

Hence verified that 2/3 is a root

For root -1/2

$\sf\: : \implies 6{x}^{2} - x - 2 = 0$

Substituting x = -1/2

$\sf\: : \implies 6{ \left( \dfrac{ - 1}{2} \right)}^{2} - \dfrac{( - 1)}{2} - 2 = 0$

$\sf\: : \implies 6 \times \dfrac{ 1}{4}+\dfrac{ 1}{2} - 2 = 0$

$\sf\: : \implies \dfrac{ 3}{2}+\dfrac{ 1}{2} - 2 = 0$

$\sf\: : \implies \dfrac{ 3 + 1}{2} - 2 = 0$

$\sf\: : \implies \dfrac{ 4}{2} - 2 = 0$

$\sf\: : \implies 2 - 2 = 0$

$\sf\: : \implies 0= 0$

Hence root verified.

From our above conclusions, we get that -1/2 and 2/3 are the roots of the given equation.